/ X' 1/ Digitized by tine Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/advancedexerciseOOschuuoft ADVANCED EXEECISES IN PEACTIOAL PHYSICS Uontion: C. J. CLAY and SONS, €AMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. ©lagflotn: 50, WELLINGTON STREET. ILeipjig: F. A. BROCKHAUS i^eto liorft: THE MACMILLAN COMPANY. aSombag: E. SEYMOUR HALE. ^ADVANCED EXEECISES IN PRACTICAL PHYSICS BY ARTHUR SCHUSTER, Ph.D., F.R.S. LANOWORTHY PROFESSOR OF PHYSICS AND DIRECTOR OF THE PHYSICAL LABORATORIES IN THE OWENS COLLEGE, MANCHESTER, AND CHAELES H. LEES, D.Sc. LECTURER ON PHYSICS AND ASSISTANT DIRECTOR OF THE PHYSICAL LABORATORIES IN THE 0WKN8 COLLEGE, MANCHESTER. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1901 [All Rights reserved] CambrilJgc : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE. nnHIS volume is intended for students who, having obtained -■- an elementary knowledge of experimental work in Physics, desire to become acquainted with the principles and methods of accurate measurement. The large and increasing number of students, who have to be taught simultaneously in a Physical Laboratory, renders it necessary that the instructions supplied should be fairly complete ; and that the exercises should be of such a nature as to enable the teachers easily to check the accuracy of the results obtained. The exercises described in this volume have been worked through by several hundred students of the Owens College who were preparing for the ordinary degree of B.Sc, and the experience thus gained has been utilised to improve the descriptions and methods adopted. It is hoped therefore that the volume will also prove useful in other laboratories. We have not aimed at completeness, being convinced that a student learns more by carefully working through a few selected and typical exercises, than by hurrying through ^ large number, which are often but slight modifications of each other. The guiding principle we have adopted in our teaching htis been to attach greater importance to neat and accurate work, properly recorded, than to the number of experiments which a student performs. All Note-books are carefully kept, no slovenly work is allowed to pass, and each exercise is repeated until satisfactory results have been obtained. VI PREFACE A student will naturally devote the greater portion of the time spent in the laboratory to measurements and quantitative work, but qualitative experiments should not be excluded. In certain parts of the subject, as for instance in Physical Optics, the educational value of setting up the apparatus and observing the general character of the effects produced is considerable, and such observations form a very useful complement to the quantitative exercises given in this book. We have endeavoured to confine the apparatus required to that commonly found in laboratories. It is not necessary that the instruments used should be identical with those described. Students should be able to introduce the slight modifications in the manipulation rendered necessary by some small differences in the apparatus. Where the differences are likely to be material, detailed descriptions have usually been omitted, and in such cases a written explanation should be supplied to the student with the instrument to be used. CONTENTS. BOOK I. PRELIMINARY. SECTION PAGE 1 T. Errors of Observation II. Meiisurement of Length III. Measurement of Intervals of Time IV. Calibration of a Spirit-Level V. Calibration of a Graduated Tube BOOK XL MECHANICS AND GENERAL PHYSICS. VI. The Balance VII. Accurate Weighing with the Balance VIII. Determination of the Density of a Solid . IX. Determination of the Density of a Liquid X. Moments of Inertia XI. E.xix;ri mental Determination of Moments of Inertia 71 XII. The Compound Pendulum XIII. Experimental Determination of the Equivalent Simple Pendulum XIV. Determination of Young's Modulus by the Bending of Beams XV. Modulus of Rigidity XVI. Viscosity XVII. Surface Tension 93 9 12 17 22 28 40 52 62 H7 79 82 m 89 vm CONTENTS BOOK III. HEAT. SECTION PAGE XVIII. Coefficient of Expansion of a Solid .... 97 XIX. Thennal Expansion of a Liquid .... 100 XX. Coefficient of Increase of Pressure of a Gas with Temperatui'e 104 XXI. Coefficient of Expansion of Air at Constant Pressure 108 XXII. Effect of Pressure on the Boiling Point of a Liquid 112 XXIII. Hygrometry 116 XXIV. Laws of Cooling 119 XXV. Cooling Correction in Calorimetric Measurements . 126 XXVI. Specific Heat of Quartz 132 XXVII. Latent Heat of Water 135 XXVIII. Latent Heat of Steam 137 XXIX. Heat of Solution of a Salt 140 XXX. The Mechanical Equivalent of Heat . . . .142 BOOK IV. SOUND. XXXI. Frequency of a Tuning-fork by the Syren . . 146 XXXII. The Velocity of Sound in Air and other bodies by Kundt's method 148 XXXIII. Study of Vibrations of Tuning-forks by means of Lissajous' figures 151 BOOK V. LIGHT. XXXIV. Measurement of Angles by the Optical Method . 154 XXXV. The Sextant 162 XXXVI. Curvatures and Powers of Lenses . . . .167 XXXVII. Determination of the Index of Refraction of a Liquid by Total Reflection 173 CONTENTS IX SECTION PAGE XXXVIII. Magnifying Powers of Instruments .... 176 XXXIX. Adjustments and Use of the Spectroscope . . 178 XL. Reduction of Spectroscopic Measurements to an Absolute Scale 187 XL I. The Spectrometer and its Adjustments . . . 194 XLII. Determination of the Refractive Index of a Solid by the Spectrometer 203 XLIII. Determination of the Index of Refraction of a Liquid. Specific Refractive Powers ..... 209 XLIV. Photometry 213 XLV. Interference of Light — the Biprism .... 216 XLVI. Meivsurenient of Wave-length of Light by the Dif- fraction Grating 220 XLVI I. Rotation of Plane of Polarisation. I . . . 224 XLVIII. Rotation of Plane of Polarisation. II . . . 227 BOOK VI. MAGNETISM AND ELECTRICITY. XLIX. The Measurement of Magnetic Fields . 232 L. Magnetic Siu-vey of the Laboratory .... 239 LI. The Tangent Galvanometer and other Current Meters 242 LI I. Comparison of Current Meters 248 LIII. Verification and Application of Ohm's Law 253 LIV. Arrangement of Cells 268 LV. The Water Voltameter 262 LVI. Standardising Current Meters by the Copi)er N'oltji- meter 267 LVII. On Mirror Galvanometers and their Adjustments . 272 LVIII. Measurement of Resistance. The Post-Office Form of Wheatstone Bridge 286 LIX. Measurement of a High Resistance .... 293 LX. Measurement of a Low RoNi.^tance . . 298 LXI. Meiisurcmont of the Resistjuico of a Galvan«»unt»r . :iiM» LXII. MeaMurcment of the Re-sistance of a Cell . 304 LXIII. Measurement of Resistance by Carey Foster's Method ."107 CONTENTS BBCnON LXIV. Change of Electrical Resistance with Temperature LXV. Resistance of Electrolytes .... LXVI. Construction of Clark Cell .... LXVII. Comparison of Electromotive Forces . LXVI II. Potentiometer Method of Measuring Currents LXIX. Thermo-electric Circuits .... LXX. Mea8iu*ement of the Mechanical Equivalent of by the Electric Method ... LXXI. Induction of Electric Currents . LXXII. Determination of the Inductance of a Coil LXX 1 1 1. Leakage and Absorption in Condensers . LXX IV. Comparison and Use of Condensers . LXXV. Determination of the Capacity of a Condenser Heat PAGE 310 315 319 323 327 329 332 335 340 350 354 356 ERRATA. . 12, last line, for " minute " read " second." 19, 3rd line from bottom, for "terms" read "turns." 73, 20th line from top, after "distance apart." add "To do this place the vertical wire of the stand provided close to one edge of the block when it is in equilibrium. Set tbe block oscillating about a vertical axis and note the time required for say 30 or 40 passages of the edge of the block past the wire in the same direction." 73, in table at foot of page multiply each number in the 6th and 7th columns by 10. 74, line 11 from top, for " 7-5 " read " 7-45," and correct rest of line to correspond. 74, line 12 from top, for "5*5" read "5-05," and correct rest of line to correspond. 87, line 15 from top, after Section III. add " see correction to p. 73." 141, in table at foot of page, for " 14-05 " read "16*05"; for " NHCl" read " NH^Cl." BOOK I. PRELIMINARY. SECTION I. ERRORS OF OBSERVATION. Our senses and judgment may be trusted up to certain limits, beyond which they begin to be subject to errors. Thus if we wish to measure a length of say 5 centimetres by means of a millimetre scale, no one will feel any difficulty in obtaining a result accurate to a millimetre or to half that quantity. But as soon as we wish to push the accuracy much further, even the most experienced observer will find the estimation difficult, and his measurement may be wrong by a quantity which is called an " error of observation." If he repeats the observations a great many times he will obtain a number of different results, which will group themselves round their average or mean value in a manner which will always shew a certain regularity, if the number of observations is sufficiently large. The study of the law of distribution of errors is of importance because it allows us to form an estimate of the accuracy with which under given conditions the measurements can be made. If there is no bias, which will cause the observation to err more often in one direction than in the other, common sense is sufficient to tell us that the arithmetical mean of a number of observations will give us the most probable result. And common sense will 8. p. 1 2 PRELIMINARY also allow us to form a rough estimate as to the limits within which the result may be trusted to be right. Suppose for instance a certain observation three times repeated has given the numbers 31, 3*3, 34, and in another case the three ob- servations have been I'l, 1'5, 7*2. In both instances the most probable value, being the arithmetical mean, is the same, viz. 327, but in the first case the observer may conclude with some confidence that his result is right to within ten per cent., i.e. the actual value will lie between 3 and 3'5, while in the second case he will attach little value to the mean obtained from such discordant measurements. Common sense like the sense of sight or of hearing may be trusted up to certain limits, and just as we can increase the efficiency of our ordinary senses by suitable instruments, so may we increase the efficiency of this common sense by an instrument which in this case is the theory of probability. To apply that theory we must in the first instance study the laws of grouping of errors, and this is best done by a graphical method. Let the curved line in Fig. 1 have the property that if N represents the number of observations supposed to be very large, the area EFHK will be a measure of the ratio njN, where n is the number of observations which shew an error greater than OH and smaller than OK. It follows of course that the unit of length chosen is such that the total area included between the curve and the line PQ is unity. It is found that in all cases which it is necessary to consider here, the curve has the same shape and may be represented analytically by the equation h Vtt r-h^X"^ I ERRORS OF OBSERVATION 3 where x is the " eiTor," i.e. the deviation of an observation from the arithmetical mean. It is seen that different cases can only differ owing to a difference in the value of h, and Fig. 2 gives Fig. 2. the curves for three different values of h. Inspection of these curves shews that the greater the value of h, the steeper the curve in the neighbourhood of the central ordinate OAy and this means that the observations are grouped more closely round their average value. We might therefore take h to be the measure of the precision of our observations, but it is usual to choose for this purpose another quantity which we proceed to define. In Fig. 1 draw two lines LM and L'M' at equal distances from OA such that the area included between these lines the curve and the horizontal axis is equal to half unit area. By the definition of the curve this means that half the total observations shew errors numerically smaller than OL. The quantity OL is called the " probable " error, meaning that errors larger and smaller than that quantity are equally probable. The probable en-or (r) may be calculated from the equation to the curves in terms of h and is found to be given by hr = -4769. The probable error varies therefore inversely as h, and the smaller the probable error the more confidence may we have in our result. The quantity which interests us most, however, is not the probable eiror of an observation but the probable eiTor of the result which is obtained by taking the mean of all the observa- tions. Assuming the curve which has been given to represent 1—2 4 PRELIMINARY I correctly the distribution of errors, the probable error of the mean of N observations can be proved to be r/^N if r is the probable error of a single observation. This result is impor- tant, for it shews that by taking the mean of four observations we double the accuracy. It is seen that we may by repeating the observation easily double or even treble the reliability of the result, but that a very large number of observations have to be taken if we aim at a materially greater improvement. The curve which represents the distribution of errors in a series of figures which do not represent single observations, but each of which already represents the mean of a certain number of them, has the same form as that already described for the single observations. The probable error of the final mean again represents an error such that smaller and greater errors are equally probable. By finding the total area which lies on the right-hand side of such an ordinate as FK (Fig. 1) we find the probability of an error greater than OK, or in other words, we find the ratio n/N, where N is the total number and n the number of errors greater than OK. It may be proved in this way that an error equal or greater than 2r, i.e. double the probable error, will happen on the average in one out of every five cases, and may therefore be said to occur fre- quently. An error exceeding 3r will only occur once in 23 cases on the average, while an error of or or more will only occur once in 1300 cases. We may say that we have a reason- able security though no absolute certainty that the result will not be affected by an error greater than four times the magni- tude of the probable error, the chance of greater error being in that case less than one per cent. It follows that one must take the mean of 16 observations in order to be reasonably certain that the error of the result shall not exceed the probable error of a single observation. The probable error for a given set of measurements might be obtained by dividing all the observations into two groups, according to the magnitude of the error. If the first group contains all observations with errors smaller than r and if by trial r is determined so that the two groups contain an equal number of observations, r is the probable error. But this I ERRORS OF OBSERVATION 5 method can only be applied if a very large number of ob- servations is available. Having again recourse to the theory of probability the following method of calculation is found to give the correct result when the number of observations is large, and will in all cases give the best result obtainable. L.et Vi, Va...v„' represent the errors, i.e. the difference between the individual observations and their mean value. Then if the probable error of a single observation is given by and the probable error of their mean is Vn \ n.n — 1 n being the number of observations. Sometimes the quantity to be determined by an experiment is not that which is directly measured, but is deduced by calcu- lation from the measurement. We must then be clear as to the error produced in the result by a certain error in the measurement. A simple case of this kind occurs if we wish to determine the area of a circle by measuring its diameter. If the diameter is D the area A is known to be JttZ^, but if an error of observation d has been committed so that the measured diameter was found to be Z) + d, the calculated area would be iir (D + dy or ^ + a = i7ri>(l + 2j+g), where a is the error in the measured area. If d is small so that the scpiare of d/D may be neglected, we find A' D' If an error d has been committed in a measurement D, it is usual to call the quantity lOOd/D the percentage error. We may similarly call d/D the "per unit error" or better, the 6 PRELIMINARY I "fractional error." The fractional error of the calculated area is therefore twice the fractional error of the measured diameter. More generally if a? be a measured quantity and y = ic*", the fractional error of y will be n times the fractional error of x. Hence the importance in such determinations as the coefficient of viscosity (Section XVI) of measuring as accurately as possible the radius of the tube, the fourth power of which enters into the calculation. As a further example we may take the measurement of an electric current by a tangent galvanometer. If the angle of deflection is 6 and the current C, the theory of the instrument gives the relation C = K tan 6, where k is the constant of the instru m ent. If an error 5 is committed in the measurement of 6, and c be the resulting error in the calculated value of G, we should have C + c = /c tan (^ H- 8) _ tan 6 + tan 8 1 — tan 6 tan 8 * Since tan S is small we may neglect its square and sub- stitute h for its tangent, the above expression then transforms easily into C +c = K tan 6 + K^ sec^ 6, .'. c = kB sec" 0. The fractional error of the calculated current will therefore always be larger than twice the fractional error of the measured angle because sin 2^ will always be less than one. The error will be smallest when 6 = 4?5°, hence there is an advantage when measuring a current by a tangent galvanometer in select- ing the instrument so that the angle of deflection is nearly 45°. More difficult still are the cases in which there are two or more unknown quantities, connected by known relations. A simple case of this kind occurs when we wish to measure the time of vibration of a magnet, for instance by observing the I ERRORS OF OBSERVATION 7 times of passage through the position of equilibrium. If t is the time of first passage and we only observe passages in the same direction, the successive recurrences of the event take place at times t, t-\-T, t + 2T, etc. The observed times are ti, to, U, &c., and if the errors are Vi, v^, v^, &c. we have the following equations ti + vo = T4-r, /3 + V3=T+2r, tn-^Vn = T + {n-\)T. These equations contain the two unknown quantities t and T and the n unknown values of the errors. There are therefore two more unknown quantities than equations. Any assumed value of T and T might be made to fit into the equations if there is no limitation to the magnitude of the error. The most probable of all possible values of r and T, are those which give that distribution of errors v which we have discussed and repre- sented by the curve in Fig. 1. The theory of probability shews that this is equivalent to saying that the most probable values of T and T are those which give the smallest values for S, where The method of calculation which allows us to determine the unknown quantities under the condition that S shall be a minimum is called the "Method of least squares." As there will be no occasion to reduce any of the observations described in this volume by this method, we need not enter into a fuller discussion, but students interested in the subject are referred to Merriman's Method of Least Squares. Errors of observation, which may be eliminated by taking a large number of measurements, are comparatively easy to deal with, but the experimentalist has also to guard against the more serious danger of being misled by "systematic errors" or errors which always affect the result in the same direction. These errors may be due to a faulty arrangement of the experi- ment, to defective instruments and also to a bias of judgment which causes the observer to commit errors which all tend to 8 PRELIMINARY I lie on the same side of the correct value. If observations are subject to errors of the last kind they are said to be affected by a "personal equation." For instance, if a star is observed as it traverses the field of view of a transit instrument, and an observer is asked to make some sign at the moment the star passes behind a wire fixed in the instrument, it has been ascer- tained that most observers will signal the passage before it has actually occurred. With practised observers the difference in time between the actual and observed transit is always nearly the same, and is called the personal equation of the observer. Personal equations of small amount probably exist in many kinds of observation, such as the reading of a Vernier. It would follow that in such cases the symmetrical curve of Fig. 1 only represents approximately the distribution of errors. But personal equations need only be taken into account in cases of extreme refinement of measurement and are only mentioned here in order to point out the existence of errors of observation which are not eliminated by the usual method of multiplying measurements. SECTION 11. MEASUREMENT OF LENGTH. The simplest method of measuring a length consists in the direct comparison of the length to be measured with a scale which serves as a standard, and which is subdivided into intervals of say 1 mm. If the length to be measured is not exactly a multiple of a millimetre, it becomes necessary in some way to measure or otherwise to estimate the fraction of a subdivision. The application of a " Vernier " to this purpose is explained in elementary books (see also Schuster and Lees's Intermediate Physics) ; the Vernier can however often be dispensed with and the subdivision estimated with sufficient accuracy by the eye. The following hints as to estimation will be found of use, but the student will only obtain confidence in his judgment by constant practice. He should not attempt to estimate more closely than to a tenth of a division, though an experienced observer will under favourable circumstances estimate correctly to a fiftieth. The eye subdivides easily an interval into two equal parts, and if the point to be measured is, e.g., at C (Fig. 3), between the scale divisions a C 3 marked 2 and 3, etren an unpractised ob- p. 3 server will put it down at once as 2*5. Should the point C lie as in Fig. 4 he 1 • ' ' ' will in his mind fix a point C at the same distance from C as that point is from '*^' the nearest scale division, then a point C" again at the same 10 PRELIMINARY II distance from C. By this means he will see at once that G is less than a third but more than a fourth part of the way from the division 2 towards the division 3 ; hence he will read between 2'25 and 2*33 and there is no difficulty in putting it down as 2'3. In Fig. 5 an observer would see if he divided the interval between 2 and 3 into two equal parts at B, and . the first half again into equal parts at B\ that 2 CB* B 3 the point G was rather nearer to 2 than to B and he would put it down as 2'2. ^^' The greatest liability to error occurs when the point G lies between '1 and '2 of an interval, and there will probably be a tendency to over-estimate the distance from the nearest division. In order to increase the power of judgment in this case, it is advisable for the student to draw two lines J. .and B and a third G, so that AG \ \ I is the tenth part of AB (Fig. 6). This ^^ . ^ should be done with AB varying in ^^' length between "5 and 3 cms. If a student once has got a good idea of a subdivision into ten parts, the estimation of subdivisions in general will not present much difficulty. Exercise. Estimate the distances of the middle lines from the left-hand lines in terms of the distance between the end lines taken as unity, in the cases shewn on the card provided. When these estimates have been entered in the note-book, measure the distances on a millimetre scale and check your estimates. When in the following exercises no special directions are given as to the method to be used in determining a length, a scale graduated in cms. and mms. should be applied to the length to be measured, adjusted to read an integral number of cms. at one end of the length, and the point on the scale corre- sponding to the other end, read. If this end coincides with a mm. division of the scale, the distance is read off immediately. If not, its position between two consecutive divisions must be estimated as explained above. II MEASUREMENT OF LENGTH 11 When the length to be measured is small, a microscope provided with a divided transparent scale in the focal plane of the eye-piece is used. If, as for instance in the viscosity exercise (Section XVI), the radius of a narrow tube is to be measured, the tube is placed vertical under the microscope and the diameter read off directly in terms of divisions of the eye-piece. The value of the scale division must be determined by placing a transparent millimetre scale under the microscope, focussing and comparing it with the eye-piece scale. The divided scale of the eye-piece may be replaced by a " Micrometer," i.e. a cross wire which can be moved sideways by means of a fine screw the turns of which are read off on the screw head. There are several forms of micrometer which need not be described as the student will have no difficulty in using a micrometer eye-piece, should he meet with one fitted to an instrument. SECTION III. MEASUREMENT OF INTERVALS OF TIME. All time measurements in a Physical Laboratory are re- ferred to some standard clock, watch, or chronometer, the error of which must be known in the few instances in which absolute time is required. In the great majority of cases intervals of time only are measured, and the rate only of the clock is required. A clock or watch is generally available which goes correctly to within a minute a day, i.e. one minute in 1440. Such an instrument would therefore allow us to determine intervals of time to within less than one part in a thousand, an accuracy sufiBcient for many purposes. When greater accu- racy is aimed at, or when the error of the timepiece is too great, the rate must be determined by comparison with some better instrument or by direct observations of the sun or a star with a sextant or transit instrument. The method to be adopted in measuring an interval of time varies with the nature of the interval, but in most cases which occur in the laboratory the interval is that between two con- secutive occurrences of some periodic event, e.g. the passage of a pendulum through its position of equilibrium. The simplest course to pursue is then to count the number of occurrences in a given tii^ie, say one or two or more minutes, according to the accuracy required. The number of occurrences divided by the time elapsed between the first and last gives the required interval. If the occurrences follow each other rapidly, as e.g. when they take place four times a minute, it requires a little practice Ill MEASUREMENT OF TIME 13 to count them. This is most easily accomplished by taking them in groups of four, counting thus: one-two-three-four, one-two-three-four, etc., stress being laid on the four in pro- nouncing, and a mark being made simultaneously with a pencil on a sheet of paper. Four times the number of marks made on the paper in a given time is the number of occurrences duiing the time. If the peiiodic time exceeds ten seconds, the total time required for a given number of occurrences is great, and the accuracy of the determination of the interval may be increased without unnecessarily increasing the time occupied, by ob- serving separately the times of a number of occurrences. In this case a watch or clock beating seconds or half-seconds is required. The observer begins a few seconds before the occur- rence to count seconds in time with the clock, while he watches for the occurrence, the counting being done thus : — twenty-o?ie, twenty-^i^o, twenty -three, the twenty being spoken lightly, and stress being laid on the ttvOy three, etc., each of which should be pronounced in coincidence with the tick of the clock. By exercising a little care the twenty may be pronounced just at the half-second, and the exact time of the occurrence fixed to less than half a second. The times of consecutive occurrences numbered 1, 2, 3, 4, 5 etc. are found in this way, and written down. To shew how the observations are reduced we take as example the determination of a time of vibration, 12 successive passages for instance of a galvanometer needle through its position of equilibrium in the same direction being observed. Let the observed times be : — Event No. Time Event No. Time 1 lib 23- 8- • 7 ll'»24» 1' 2 12 8 11 3 22 9 20 4 32 10 30 5 41 11 39 6 51 12 49 14 PRELIMINARY III If we only make use of the first and last observations we should find the interval of time for 11 vibrations to have been 1" 46', and therefore the time of a single vibration to have been 9^-64. The intermediate observations are not made use of in this method of calculation, and there is therefore some loss of accu- racy in the final result. To discover the method of reduction which, without too much numerical labour, should give the best result, we notice in the first instance that the probable eiTor in the measurement of a time interval will be the same, whether that interval be large or small. If e is that probable error, and if the measured interval includes n complete vibra- tions, the probable error of the periodic time calculated from the n intervals will be ejn. If instead of observing the time of n vibrations we had observed the time of a single vibration, and repeated the ob- servation n times, the mean of the results so obtained would have given a probable error ej^Jn (Section I). As ejn is smaller than e/Vw in the ratio of Vn : 1, it follows that a better result is obtained if only two observations are taken, viz. one at the beginning and one at the end of n consecutive intervals, than if n separate intervals implying 2n observations are taken. This shews that it is not only the number of observations which determines the accuracy of a result, but their intelligent ar- rangement and reduction. Returning to the above example it is easily seen that the result may be improved by making use of the two first and two last observations. The interval between the first and the 11th is 96* for 10 vibrations or 9'60 for one, the interval between the second and last is 97* for 10 vibrations or 9-70 for one, the mean being 9*65. The probable error of each of these results separately is e/10 and that of their mean therefore e/10 V2 or nearly e/14, as compared with e/11 in case only the first and last obser- vations are made use of. More generally taking the first p and the last p observa- tions and taking the difference between the {n —p + l)th and the first, that between the {n—p-\- 2)th and the second, and so on until we come to the difference between the pih. and Ill MEASUREMENT OF TIME 15 last observation, we secure p values of n — p intervals, the time of vibration calculated from each separately will have a pro- bable error el{n—p) and the probable eiTor of their mean will be e/(n — p) y/p. The smallest probable error is obtained when (n —p) hjp is as large as possible for a given value of n, and it is easily shewn that this is the case when p — w/3. Appl}dng this to the above example it follows that it would be best to reject the 5th, 6th and 7th observations and to arrange the rest as follows: — Event No. Time Event No. Time Time of 8 intervals 1 2 3 4 11»>23'°3« 12 22 32 9 10 11 12 1 1^ 24°* 20" 30 39 49 jm 17. 18 17 17 Sum 5- 9- Mean 1 17-25 .*. Interval 9-656 The observations which are not made use of need not be written down at all, but in that case great care is needed to avoid making an error in the number of observations which are omitted. The liability to error is reduced, and the re- liability of the result not materially interfered with by a slightly different arrangement. If 'p intervals are observed, i.e. jo + 1 observations taken, then p intervals omitted, and p again observed, the time when the last set of p observa- tions should commence may be readily calculated. Thus in the above example if four intervals had been measured the diflference of the first and fifth observation would have given 38' for the time of four intervals, adding 38" to the fifth obser- vation the calculated time for the ninth would have been llh 23™ 41' -I- 38' = 11'* 24°* 19', the observer could therefore have rested till near that time and then carefully watched for the occurrence, which as the table shews actually took place at 24™ 20-. 16 PRELIMINARY III The method which has been explained in connection with the measurement of time intervals, is equally applicable in other cases, such as that of temperatures measured with a uni- formly rising or falling thermometer read off at equal intervals of time, the rate of fall or rise of the thermometer being the quantity to be determined (Section XXIV). Sections (XIX) and (XX) also shew examples of the same method of reduction. SECTION IV. CALIBRATION OF A SPIRIT LEVEL. Apparatus required : Spirit Level, hoard with levelling screiv, screw gauge. The Spirit Level consists of a slightly bent tube (Fig. 7) partly filled with alcohol. The bubble of air or of vapour of alcohol left in the tube will al- ways set itself so that it is at the highest point of the tube. The tube is generally divided as shewn in Fig. 8, so that it is easily seen when the bubble is in the centre of the tube. If the level is in proper adjustment the base of the plate to which the level is attached should be hori- zontal when the bubble is in the central position. It is therefore necessary that the central point D of the tube (Fig. 7) should be farthest from the supporting plate AB. An adjusting screw T is generally provided on one side of the level, by means of which the tube can be tilted with respect to the base to secure that this shall be the case. If the table on which the level is placed is nearly but not quite horizontal, the bubble may not stand in the middle, but be slightly displaced ; the position of its centre may be read off in scale divisions, and if the value of the scale division is known, the inclination of the table to the horizontal may be calculated. It is the object of the present exercise to calibrate the divisions of a level so as to find their values in I I ft.i.|iiMHc =^|m.|iiio loj Figs. 7, 8. 8. P. 18 PRELIMINARY IV angular measure, and also to shew how even if the spirit level is not itself in proper adjustment it may be used to determine whether a surface is horizontal. If the level has not been tested and when placed on a table shews the centre of its bubble shifted to one side, it is un- certain whether this is due to the fact that the level is not in adjustment or that the table is not horizontal. But if the level be turned through 180° round a vertical axis so that the points A and B are interchanged, we may find out which is the correct interpretation. If the table is horizontal and the level wrong, the bubble should remain at the same place, for that point of the tube which is farthest from AB will remain so in whatever position the level is placed. If on the other hand the level is correct and the table inclined, the bubble will move over to the other side and its new position will be changed with reference to the level, but will be the same in space as previously. You are provided with a board supported on three legs, one of which can be screwed up or down so as to alter the angle of inclination of the board to the horizontal. In order to be able to measure the amount through which the board is tilted, the movable leg carries a divided circle in the manner of a* spherometer screw. The rotation of the screw is read on the divided head and its pitch must be separately de- termined. The board is to be placed with its legs on thick ^ Fig. 9. glass plates, and the Spirit Level placed on the marked space at the centre of the board, with its length perpendicular to the line joining the fixed legs of the board. One end of the level is provided with an adjusting screw. Take a note of the side on which the adjusting screw is IV CALIBRATION OF A SPIRIT LEVEL 19 placed. Let it be for the sake of uniformity the right-hand side. If the level has no adjusting screw, mark one end in some way and place it to the right. Level the board by means of its screw till the bubble is nearly in the centre. Let the readings of the ends of the bubble be, e.g., — *4 and 4- '5, the divisions at the screw end of the level being called positive, those at the other end negative. Now remove the level and reverse it so that the screw end is to the left. If the reading of the screw end of the bubble remains 4- "5, the board is horizontal. If the reading is altered to + '6 say, take out half the error by means of the board screw, so that the end of the bubble now reads + '55. Reverse the level again so that its screw is on the right hand. This reversal should not alter the reading of the screw end of the bubble from 55. If it does, again take out half the error by means of the board screw, and reverse again. In this way a position of the board screw is found, such that reversal of the level produces no change in its readings. The board is then horizontal. Adjust the screw of the level till, when the level is on the board, the two ends of the bubble are at equal distances from the centre of the scale. The level is then in proper adjustment for use. If the level provided has no adjusting screw, the error of the level is thus determined and the calibration may be pro- ceeded with. To determine the value in degrees, minutes, and seconds, of each division of the level, place it with its screw end to the right. Adjust the board screw till the centre of the bubble is at 0, and read the screw head. Now, adjust the board screw till the centre of the bubble is at 1 to the right of zero, and read the screw head, recording the number of terms and decimals of a turn. Repeat, placing the centre of the bubble at 2, and so on for the other divisions of the right-hand scale. 2—2 20 PRELIMINARY IV Again adjust the centre of the bubble to 0, read the screw head, and take observations as the bubble is moved to the left. Determine the pitch of the screw by first adjusting the screw till the centre of the bubble is at zero, reading the screw head, then inserting under the screw the small piece of plate- glass 5 mms. thick provided, again adjusting the screw till the bubble is at zero, then reading the head and recording the number of turns and parts necessary to make the adjustment. Determine by the screw gauge the thickness of the glass-plate, and thence calculate the pitch of the board screw. Measure the perpendicular distance / from the screw leg to the line joining the other two legs of the board. Then if n = number of turns of board screw from its reading when the bubble was at 0, and p = pitch of the screw, np = distance through which screw has been raised, and np/'l = angle in circular measure through which the board has been titled. (Since the angle is small.) Or in angular measure . 180° np TT Determine in this way the angle of tilt for each observation, and draw up a table as follows : — Spirit Level A . Reading of centre of bubble Board screw d 5^ on screw end of scale head •100 1 1-350 9' 2" 9' 2" 2 2-650 18' 25" 9' 23" 3 3-783 26' 36" 8' 11" 4 4-567 32' 11" 5' 45" 5 6-183 36' 42" 4' 31" 6 5-683 40' 19" 3' 37" Similarly for readings of the bubble on the other end of the scale. IV CALIBRATION OF A SPIRIT LEVEL 21 The column headed 86 gives the angle through which the level has to be turned in order to shift the bubble through one division. If the curvature of the tube of the level is cx>nstant, so that its axis is a circle, the numbers in the last column should be the same. The particular level to which the above numbers refer had a greater curvature nearer the end than at the centre. Students may find that the length of the bubble is not constant owing to the evaporation or condensation of the alcohol due to changes of temperature. Hence the necessity for reading both ends of the bubble. SECTION V. CALIBRATION OF A GRADUATED TUBE. Apparatus required : Graduated tube, mirror^ lens to he used as Tnagnifying -glass, and clean mercury. The graduations on a tube which is to be used for measurements, are as a rule placed at equal distances apart along the tube, but if the tube is used to measure volumes, the assumption that distances between the graduations repre- sent equal volumes of the tube, will lead to error unless the tube is of uniform bore. It is the object of calibration to find the correction which must be applied to the reading on the scale of equal lengths, in order to convert it into one of equal volumes. The calibration correction is that part of a division which must be added to any reading on the equal length scale in order to obtain the reading of the same point on an equal volume scale. Let a tube be provided with equidistant graduations, 1, 2, 3, &c., see Fig. 10, and imagine graduations I., ii., ill., &c., along the same tube, such that the intervals between each represent not equal lengths, but equal volumes of the bore of the tube. 1 1 1 2 3 « 1 1 1 a 1 t 1 1 1 1' 1 D nr 11 Fig. 10. 1 V 1 Thus the zeros of the scales being supposed to coincide, the division 1 corresponds to rather more than I. on the equal volume scale line, the correction at the point 1 is therefore positive. Similarly the corrections at points 2 and 3 are positive, and at 4 negative. V CALIBRATION OF A GRADUATED TUBE 23 Let the corrections at successive divisions be Xq, a?,, x^, a?,, &c., and suppose a mercury thread, which if the temperature is constant will be of constant volume, say about equal to that of a scale division, to be pushed along the tube. In Fig. 10 the ends of the thread lie approximately at the points marked 3 and 4. Let 84* be the excess of the length of the thread over the distance between the points 3 and 4. The point 3 of the equal length scale would read 3 + .^3 on the equal volume scale, the point 4 would read 4s + x^, the true distance between these points would be therefore, not 1 but 1 +x^ — Xi, and if a be the number of divisions between the points 3 and 4 on the equal length scale, the subdivisions being left out in the figure, the true length of the interval would be not a but o + a;4 — x^. The excess ^4' read on the equal length scale being small, will not be sensibly different when read on the equal volume scale, hence the coirected length of the thread becomes a 4- ^4 — a:, + 84' and this we know must be equal to a constant, say I, wherever the thread is placed. Applying this equation successively to the different positions of the thread we obtain the following n equations ^'1 — Xf, -\- hi'' = I — a x^ — Xi'\'h^ = l~-a a?n-a:n_i + 5n*»-' = ^-a (1) involving n-\-2 unknown quantities, a?©. ^i,---^n, ^' Hence we must know two of the corrections, or at any rate two relations between them, to solve the equations. An important case of calibration is that of a thermometer between freezing point and boiling point, the corrections at these two points being determined by experiment, i.e., x^ and ar^ are known, and there only remain n unknown quantities. In that case, to solve the above set of equations, add up all the equations, and write B for the arithmetical mean of all the S's. The addition gives Xn - x^f -\- nB ss n {I ~ a) (2). 24 PRELIMINARY V From which, since Xn and x^ are given, I — a can be calculated. soi may then be calculated from the first of the equations (1), since ^" has been determined. Having found oo^, x^ is calculated from the next equation, and so on successively for all the corrections. A tube is given to you divided into millimetres (Fig. 11). It n Fig. 11. is to be calibrated between points 1 6 cms. apart, the errors being required for points at a distance of 2 cms. apart. That is to say, assuming the first and seventeenth centimetre division to be correct, the errors at the third, fifth, &c. division are to be found. The most difficult part of the operation consists in obtaining a mercury thread of the required length. It may most easily be accomplished by slipping a short piece of indiarubber tubing over one end of the glass tube. The free end of the rubber tube is then compressed between the finger and thumb of the right hand, so as to close it completely, the end of the glass capillary tube dipped into the mercury, and then the rest of the rubber tube compressed between the finger and thumb of the left hand to expel a little air from it. On releasing the left hand, a thread of mercury is drawn up. The capillary tube should then be quickly placed nearly horizontal with the lower end over the mercury in the bottle, and the end of the rubber tube released. By gently tilting the tube, the thread may be made to approach the lower end of the tube, and a small drop of mercury forced out and cut off with the finger-nail or a knife. By carrying out this operation several times, the thread maybe diminished in length till it occupies between 19 and 21 of the small scale divisions, i.e. within 1 mm. of 2 centimetres. When this is the case, move the thread till its left-hand V CAUBRATIOX OF A GRADUATED TUBE 25 end nearly coincides with the division on the left at which the calibration is to commence. Place the tube on a strip of mirror glass and read both ends of the mercury thread by means of a magnifying lens, avoiding parallax by placing the eye so that its image is covered by the part read and then slipping the lens into position. The excess of the observed length of the thread over 20 scale divisions is the 81* of the preceding equations. Now move the thread forward 20 small scale divisions so that its left-hand end nearly occupies the position of the right- hand end in the previous case, and obtain 8^^ as before. Continue the operation till the ends have been read in eight successive positions, and then take readings as the thread is moved backwards. The means of the B's observed at each part of the scale going and returning should be used in the calculations, which should be carried out and tabulated as shewn below. As in this exercise we only desire to subdivide the distance between two given points of the tube into divisions of equal volume, we may take the corrections at the initial and final points as zero. Putting Xq = x^ = 0, equation (2) becomes / — a = 5. Hence from (1) Xi — 8 — 81^, x^--x, + 5 - 8,» = (5 - V) -f- (3 - S,»X -r,=:a:2+ 3- 53»=(S- V) -f (s -a,o -H (a - 8,% = 0. In this case x^ is zero, and this will furnish a check on the arithmetical calculations. A check for some of the intermediate points may be obtained by taking threads 4 and 8 cms. long. Calling the first division 0, the 4 cms. thread will give the corrections at the points 4, 8 and 12, while the 8 cms. thread will only give the correction at the division 8. To calculate the corrections it is necessary first to find 3, which is the mean of the quantities 8i", 8^, &c. If these are 26 PRELIMINARY tabulated as in the annexed table and the mean found, the differences 8 — Sj**, 5 — §2^ may be written down as in the fifth column and the corrections obtained by successive addition. Arrange your observations and calculations as follows, giving the mark on the label attached to the tube so that it may be identified : — Tube B. Date: 3 Jan. 1898. Going Returning Mean &c. 6-61 &c. Corrections - 04 to 2-03 = 2-07 1-98 to 4-04 = 2-06 2-06 2-06 &c. 2-03 2-02 2-01 2-01 --05 to 2-0 = 2-08 &c. 2-075 2-065 2-062 2-067 2-032 2^017 2-017 2-005 •075 •065 •062 •067 •032 •017 •017 •006 - -0325 - -0225 - -011)5 - -0245 + •OlOo + ^0255 + ^0255 + ^0375 - •033 = ^2 - •055 = ^4 - •076 = a:g - -099 = 0:8 -•089 = a:i, -•063=a;i3 - •038 = a;i4 -•000 = a;ie Sum = •340 Mean = 5= •0425 -•05to4-10 = 4-15 4-16 &c. 4-09 4-03 --06 to 4-10 = 4-16 &c. 4-155 4-160 4-095 4-005 •155 •160 •095 •005 -•041 -•056 + •009 + •099 -•051 = a;4 -•107 = ^8 - •098 = :ri2 -•00 =x,. Sum = •415 Mean = 5 = -104 -•03 to 8-12 = 8-15 7-97 -•04 to 8-12=8-16 8-155 7^955 •155 -045 -•100 + •100 - •10 = ^8 Sum = •110 Mean=5= •055 Taking the readings along the tube as abscissae and the corrections at these readings as found by the first measurements as ordinates, above or below according as the correction is positive or negative, plot a " Calibration Curve " for the tube as shewn below, Fig. 12. Do the same for the second and third sets of readings, taking the same abscissae. CALIBRATION OF A GRADUATED TUBE 27 If I is the length of the mercury thread in any position, and A the mean area of the cross section of the tube within the part occupied by the mercury thread, Al is the volume of the thread. As this volume remains constant as the thread is moved, the length of the thread varies inversely as the mean area of cross section within its length. Hence by taking again dis- tances along the tube as abscis- sae, and erecting at the different points occupied by the centre of the mercury thread ordinates inversely proportional to the observed lengths of the threads, we get a representation of the way in which the cross section of the tube varies. V^ ' 1 ' J -Sx i S> J ■' - 5^ ^ H- :_^t. _s - - -_ -t:,--- ^ I • - \ /-s- ^S t J. M.. 5^-2. _L :_ _ HB--- ^^-__L : t::: ±_ Fig. 12. Centre of Length of Reciprocal of thread thread length 1 cms. 2075 cms. •482 3 •065 •484 5 •062 •485 7 •067 •484 9 •032 •492 11 •017 •495 13 •017 •496 15 2-005 •499 Plot these reciprocals as shewn below, Fig. 13. The most important practical application of the preceding exercise occurs in the calibra- tion of thermometer tubes. All thermometers used for accurate work should be calibrated either by the maker or by the observer. Information as to the methods of breaking off a mercury thread and complete methods of cali- bration will be found in Guil- laume's Thermoniiirie, Fig. 18. |..«4»- ;:t-- pi: \ :^ii _L -,: _^4^-^ 4 ^tX.— 'Ziml. ZIhh---^-----**- HI--- BOOK II. MECHANICS AND GENERAL PHYSICS. Fig. 14. SECTION VI. THE BALANCE. Apparatus required: Delicate balance, centigram rider, two 500 gram zveights. The Balance, in its simplest form, consists of a straight beam AB (Fig. 14) provided at its centre with a knife edge G on which it is supported, and carrying at its ends the pans P and Q, on which the masses to be compared are placed. If the two halves of the beam are alike in every respect, so that the centre of gi^avity of the beam is at the point of support G, and if the pans have equal mass, the balance will be in neutral equilibrium, whether unloaded or loaded with equal masses. If unequal masses are placed in the pans, the equilibrium will be unstable, however small the inequality may be. It would be inconvenient to have a delicate balance con- structed according to this principle, for as two masses are never exactly equal, the balance would never be in equilibrium. A delicate balance, to be useful, should allow us to determine the difference between two nearly equal masses, and it is proved in treatises on Me- chanics that this can be done by constructing the beam AB of the balance so that its centre Fig. 15. VI THE BALANCE 29 of gravity is slightly below the fulcrum G. The balance will then have a stable position of rest even if the two weights P and Q are not quite equal, and the amount of inequality may be determined by an observation of this position. The pans of a delicate balance are suspended from agate planes supported on knife edges at the ends of the beam, which is itself provided at its centre with a knife edge resting on an agate plane at the top of the pillar (Fig. 15). The accuracy which can be attained in weighing, depends to a great extent on the freedom with which these three knife edges can turn on their planes. In order to save these delicate parts as much as possible from wear and tear, a so-called arrestment is provided, by means of which the central knife edge of the beam may be raised from its agate plane, while at the same time the agate planes of the pans are raised from their knife edges. The balance is enclosed in a glass case, and the aiTestment worked from the outside by the screw head in front of the case. Wlienever the balance is not in use the arrestment should be raised. The change of position of the beam of the balance is observed by the help of a pointer attached to the beam, the free end of which moves in front of a scale at the foot of the pillar. Before beginning to work, students should carefully inspect T n Fig. 16. ^ 30 MECHANICS AND GENERAL PHYSICS VI the balance (without opening the case), and make in their note- book a rough sketch of the mechanism of suspension and arrestment. The reference books in the laboratory library contain detailed descriptions of the balance, which may be consulted by the students with advantage. Exercise I. To find the zero of the unloaded balance. Carefully lower the arrestment of the balance, and watch the movement of the pointer. If it moves slowly through a range of from 3 to 6 divisions of the scale, observations may be commenced. If the extent of the swing is greater, it is a sign that the arrestment was not handled with sufficient care ; it should be gently raised again, and lowered more slowly. If the swiog is too small, open one of the windows of the balance and produce a weak current of air by gently waving your hand a few times inside the case, being careful not to bring it into contact with any part of the balance. A little practice will enable you to obtain a workable swing. The window is then shut and the observations commenced. As it would take too long to wait until the balance has come to rest, the position of rest must be determined by observations made during its motion. For this purpose the turning points of the pointer on the scale must be observed. If the balance moved without any friction, the pointer would move to equal distances on both sides of the position of rest, which could therefore easily be deduced from two successive readings of the turning points. Owing to friction the oscillations gradually diminish, and in order to find the position of rest more than two readings are required. The following example will shew how the position of rest of the balance is calculated from three observations of the turning i i I *^ I i I points of the pointer. I ■ ■ I I I M I I I I I I I I I I I I I o ^i. ^r u ^ri. 5 10 15 20 Suppose that after the ar- restment has been carefully ^^' VI THE BALANCE 31 lowered, the pointer moves to the right, then turns back again to the left at a point which is not observed. Let the three following turning points, indicated by arrows in Fig. 17, be read off as follows: ll'O, 168, 11*7, the centre of the scale over which the pointer moves being marked 10 in order to avoid negative numbers. If there were no friction the arith- metical mean between 110 and 16*8 would give the position of equilibrium. In reality this mean (13*9) gives too small a value, for without friction the needle would have moved beyond 16'8. For the same reason the mean of 16'8 and 117 (14*25) would give too large a value, and as the frictional retardation has been practically the same during the forward and during the backward swing, the errors of the two means will be equal and in opposite directions, and hence the arithmetical mean between 13'9 and 14*25, i.e. 1408, gives a sufficiently correct result for the required position of rest. The same numerical result is obtained by taking the mean of the two successive swings on one side (1185 in the above example) and then the mean between this number and the intermediate turning point on the other side. Thus the mean of 11 35 and 16 8, or 1408, would be the true position of rest. If the numbers in the first instance be read off correctly to a tenth of a division, the final result involving three readings may be given to another decimal place, for though the observations may not be sufficiently accurate to fix the final position within a htmdredth of a scale division, they should be sufficiently accurate to determine it more nearly than a tenth of a division. Practice will enable the student to read with certainty to less than a tenth of a division. The effect of small errors of observation on the final result will be diminished by taking more than three readings, care being taken, however, to take an uneven number, beginning and ending on the same side. The same method of reduction may then be employed. There is, however, a considerable loss of time when too many readings are taken, the time being better employed in taking a fresh observation altogether, and unless there are special reasons to the contrary, students in the laboratory should take only five successive readings and put 32 MECHANICS AND GENERAL PHYSICS VI them down in their note-book as in the following example, noting the temperature indicated by the thermometer in the balance case. 18 June 1896. Balance: A. Turning points left right 7-4 7-6 7-7 131 130 Sums Means 22-7 7-57 26-1 13-05 Position of rest 10-31 Temperature 18°-1 C. Whenever a weighing is taken the observations should be recorded as above, but in writing out results, the final positions of rest and the temperatures need only be given. To bring the balance to rest raise the arrestment gently so that it lifts the beam when the pointer is passing through the point of rest, and thus injures the knife edge as little as possible. Take thi-ee sets of observations of the position of rest of the balance, raising the arrestment between each set. Find the mean value of the zero as determined by the three sets. The temperature of the balance case is required in this and the next exercise, in order that it may be ascertained by observations taken on different days whether the zero of the balance shews any appreciable change depending on temperature. For ordinary purposes it is not usual or necessary to observe the temperature. VI THE BALANCE 33 Exercise II. To find the sensibility of the unloaded balance. The change produced in the position of rest of an unloaded balance by 1 milligram excess of weight on one side, is known as the "sensibility" of the unloaded balance. The sensibility varies to some extent with the load placed on the pans. In order to produce a small excess of weight (less than 1 centigram) a rider is provided, which, by means of a sliding rod, may be shifted along the beam. The weight of the rider is 1 centigram ^ but the nearer it is to the fulcrum of the balance, the smaller will be its turning moment ; each half of the beam is divided into 10 equal parts marked 1, 2... from the centre to the end, so that if the rider stands at say 2, its effect is the same as the addition of 2 milligi-ams to the weights on the pan. To find the sensibility it will in general be suflScient, by means of the rider, to add to one side a weight equivalent to 1 mgrm. (or for sake of greater accuracy 2 mgi-ms. dividing the resulting change of position of the pointer by two). In order to obtain suflScient practice in accurate weighing, the student is required to determine the positions of rest, varying the excess- of weight from 8 mgrms. on one side, to 8 mgrms. on the other of the centre of the beam. If, however, the balance is very sensitive, it may not be possible to keep the swings within the limit prescribed above (6 scale divisions), when an excess of 6 or 8 mgrms. is placed on one side. The observations should be taken and recorded as in Exercise I. The numbere obtained should bt; siinimarised in the Book of Results as follows : — » The weights of riders sold commercially often differ so much fh)m their nominal value that serious errors may be introduced if they are used without being tested. 8. P. 34 MECHANICS AND GENERAL PHYSICS 20 June 1896. Balance: A. VI Time, 10 h. 10 m. Temperature of Balance Case, l8°-2 C. Sensibility Mgs. excess of Weight Position of Rest Difference 10-07 On left side : 8 12-79 6 12-17 •62 4 11-43 •74 2 10-81 •62 On right side : 2 10-13 9-46 •681 •67/ •338 4 8-81 •65 6 8-11 •70 8 7-42 •69 10-15 1 Temperature of Balance Case, L8°-9C. Time, 10 h. 50 m. Position of rest at beginning of experiment ... 1007 „ „ end „ 1015 Temperature at beginning „ 18^-2 C. end „ 18°-9 Sensibility of balance '338 Note. — In calculating the sensibility of the balance , the only numbers in the above table taken into account, are those which give the deflections produced by 2 mgrms. on either side of the zero of the balance. The sensibility of the balance may be increased or dimi- nished by raising or lowering the centre of gravity of the beam, and each balance is provided with an adjustment for this purpose. But it must be remembered that an increase in the sensibility does not necessarily mean an increase in the accuracy with which a weighing can be made. As the balance is made more sensitive, its time of vibration increases, each weighing consequently takes a longer time, and small changes in the zero of the balance may take place owing to changes of temperature or other causes. The longer the time, the greater the proba- VI THE BALANCE 35 bility of the occurrence of such disturbances. Moreover, the extra time thus spent might just as well be spent in repeating the weighing with the former sensibility, and the result would probably be improved more in this way than in the other. There is always a limit to the accuracy with which a balance will weigh, and once that degree of accuracy has been attained by a careful reading of the position of rest, it does more harm than good to attempt to increase the sensibility. Students will obtain the best results by carefully practising the method of obtaining the position of rest which has been explained above, depending on accuracy of reading rather than on a great sensibility of their balance. Exercise III. To find the zero of the loaded balance. If two exactly equal masses were available, to tind the zero of the balance when loaded with them, it would simply be necessary to find the position of rest when the masses were placed in the pans. As however absolute equality is not easily attainable, we must find a way to determine the zero notwithstanding the inequality of the masses. If the position of rest has been found with two weights, P and Q, in the pans, then if the weights were equal, on interchanging them the position of rest would be exactly the same, and this would be the case independently of any adjustment of the balance. Even if the arms are not equally long, or if they alter their length by bending, two equal weights may still be interchanged without change in the position of rest. But suppose the position of rest is 12-3 when P is on the left-hand side, and 107 when P is on the right. If Q were slightly incrccosed, the first number would be diminished and the second number increased by equal amounts, and the difference in the two readings would diminish. It follows that if the increase in Q were such that the new position of rest is the arithmetical mean between 12*3 and 10*7, i.e. 11 "5, no change would be produced on interchanging the weights, and hence 11*5 is the position which the balance would take up if equal weights P were placed in the pans. If the sensibility of the balance were known, the experiment 3—2 86 MECHANICS AND GENERAL PHYSICS VI would give the difference between P and Q, which is evidently the weight required to be added to Q in order to produce a difference of 12*3 — 11*5 = "8 scale division. Two masses weighing nearly 500 grams are provided. Com- pare the position of rest when no weights are in the pans, with the position of rest when these equal masses are placed in them. Observe the temperature. Take one observation without weights, then one with the weight marked 1 in the left and that marked 2 in the right pan, one with the weights reversed, then one with the weights as at first, then one without weights. Again observe the temperature. Note. — If the masses are not placed quite centrally on the scale pans, the pans will oscillate, and errors may thus be introduced. In accurate weighing this should be avoided as much as possible, by arranging the weights symmetrically. Oscillations may be stopped before the arrestment is lowered, by carefully placing the open hand so as just to touch the pan with thumb and forefinger, or by touching it with a camel hair brush. The results are entered as follows : 20 June 1896. Balance: A. Temperature .. 18°-5C Zero of balance without load .. 11-34 Weight no. 1 on left pan ; position of rest . .. 1010 a ^ i> )) .. 11-42 i) ^ }} }) .. 1014 Zero of balance without load .. 11-44 Temperature .. 18°-9C. Mean temperature .. 18°-7C. Zero of balance for load 500 grs .. 10-77^ Mean zero of balance without load ... . .. 11-39 Change of zero •62 Change of temperature •4°C. 1 This number is obtained by combining the mean of the readings found with weight no. 1 on the left pan, with the reading found when that weight was on the right pan. VI THE BALANCE 37 Exercise IV. To find the sensibility of the loaded balance. In Fig. 15 the three knife edges are represented as being in the same straight line, but this is not necessarily the case. Let A and J5, Fig. 18, represent the knife edges from which the pans are suspended, and C the knife edge on which the beam of the balance rests. We shall assume that by means of the ^' movable vane attached to the beam, the centre of gravit}^ G, of the beam alone, has been adjusted so that when it is vertically beneath C the line AB i^ horizontal. Let the dis- tances of G and of the line AB below C be h and k respectively. Then if P be the weight of the pan and contents, and a the length of the arm on the left, and Q that of pan and contents, and h the arm on the right, and if 6 be the rotation of the beam produced, we have, taking moments about C, P (a cos e-k sin 0)^Q{b cos d + k sin 6) + Wh sin 6, , ^_ Pa-Qb If Q = P -\-j), then since as a rule b will be equal to a, and k will be small, the expression may be written, neglecting pk in the denominator : — The deflection 6 produced by a given excess of weight p on one side will therefore diminish with the load if k is positive, and increase with the load if k is negative. But the value of k itself will vary with the load, as the beam of the balance always bends a little, and the knife edges descend as the load is increased. The makers sometimes adjust the knife edges so that the line AB is slightly above C when the balance is unloaded, and below C when the balance is loaded with more than half the maximum load for which it is intended. In that 38 MECHANICS AXD GENERAL PHYSICS VI case the sensibility will first increase and will then decrease with the load. Theoretical Exercise for Advanced Students. — Writing in- stead of k, — k^-h A.P in order to indicate its change with the load, prove that the maximum sensitiveness will be reached k for the weight P' = ~ ; and that A^o and \ are determined by the equations 2\ (Pi - P,) (Pi + P. - 2P') = pa (cot e, - cot e,\ ko (Pi - P2) (Pi + P2 - 2P0 = P'pa (cot (9i - cot 0^), where a represents the length of the beam, and p the additional weight which produces deflections 6^ and 6^ when the loads are Pi and P2 respectively. With the help of these equations the quantities ko and \ could be calculated ; but the result depends on the assumption that the bending of the beam is symmetrical and proportional to the load, and that it does not throw the centre of gravity of the beam to one side. Determine the positions of rest of the balance under the following conditions : — (1) The two 500 gram weights in the pans. (2) An excess of 2 mgrs. on the left. (3) Asin(l). (4) An excess of 2 mgrs. on the right. (5) As in (1). Enter your results as follows : — 7 Oct 1896. Balance: A. Time at beginning of experiment, 11 h. 15 m. Position of rest (1) with load of 500 grms 7-98] (2) „ + excess of 2 mgrs. on left 8-381 8-04 '34 (3) „ ... 8I0J] (4) „ + excess of 2 mrgs. on right 7'84 >8*19 '35 (5) „ 8-28 J Time at end of experiment, 11 h. 32 m. Mean sensibility '172. VI THE BALANCE 39 Exercise V. To determine the ratio of the arms of the balance. If we assume that the ratio of the lengths of the arms of a balance is not affected by the bending of the beams, the change of zero with the load can only be due to an inequality in the arms of the balance, and the data obtained in Exercises III. and IV. are sufficient to determine that ratio. In Exercise III. the change of zero from no load to one of 500 grams was found to be '62 divisions, and in Exercise IV. the sensibility was found to be '17 ; hence to bring the loaded balance with its zero 10*77 to the unloaded zero 1139 we should have to add '62 -T- = 3*6 mgrs. on the left-hand side. Let this quantity be denoted by p, and let the lengths of the left and right arms of the balance be a and b respectively. When the loaded balance comes to rest, the moments which act on the beam will be the same as in the case of the unloaded balance, with the addition of P •¥ pa on the left side, and Pb on the right side, and if the position of equilibrium is unaltered it follows that (P + p)a = Pb, or a P oOO As the beam of the balance has a length of about 22 cms., the error of adjustment of the knife edges only amounts to •00016 cms. SECTION VII. ACCURATE WEIGHING WITH THE BALANCE. Apparatus required : Balance, piece of quartz, box of weights. The method to be adopted in weighing depends on the object for which it is carried out. Extreme accuracy always means the spending of a good deal of time on the observations, which would be wasted if from the nature of the case such accuracy were unnecessary. In chemical analysis relative weights only are required, and an inequality of the arms of the balance will not affect these so long as the weighings are carried out on the same side, and the method of Exercise I. p. 30 may therefore be adopted. More- over, in many cases the errors introduced by impurities in the substances or by other causes, may amount to several mgrms., and as it would then be absurd to conduct the weighing cor- rectly to the 10th part of a mgi-m., the method may be further shortened, by reading three turning points instead of five, by assuming the zero to remain constant during the series of weighings so that it need only be taken once, and especially by using a less sensitive balance having a shorter time of vibration. If the same balance is always used, much time may be saved by determining its sensitiveness for different loads once for all. Unless the adjustment is altered, the sensitiveness will remain the same for a considerable period. The custom of assuming the zero always to be at the centre of the scale, is not to be commended, as it may cause serious errors. The student should first learn to weigh accurately VII ACCURATE WEIGHING WITH THE BALANCE 41 irrespective of the time it takes; he will gradually learn to weigh quickly, and to know how to save time when great accuracy is not required. In all cases where absolute and not merely relative weights are required, some method should be used which eliminates all errors of the balance, and corrections must also be made for the upward pressure of the air in which the weighing is conducted. We proceed to explain the way in which this is done. Corrections to he applied to weighings for the buoyancy of the air. When a body is surrounded by air, it is acted on by an upward force equal to the weight of the air it displaces. Two bodies having equal masses but different densities, will occupy different volumes, and if these bodies are placed on the two pans of the balance, it will show an apparent inequality in the weights owing to the difference in the upward pressure of the air on the two bodies. On the other hand, if two masses of different densities balance each other completely when placed on the pans, they will not in reality be equal. If M is the mass of the substance weighed, and p its density, its volume is Mjp, and the upward force of the air will be equal to the weight of a mass MXjp of air, where X is the density of the air. Hence the resultant downward force which acts on the beam of the balance, is the same as if the surrounding air had been removed and a mass M{\ —X/p) placed in the pan. Simi- larly a mass W of density a- in the second pan, produces the same downward force as a mass W{1 — X/<r) suspended in vacuo. If the balance, supjxjsed to possess equal arms, is in equilibrium, we have : — il/(l-X/p)=F(l-X/o-). Hence if = F (1 - X/<r)/(l - X/p) = Tr(l - \/cr + X/p) approx. The last approximate result is obtained by neglecting the squares of the small quantities X/p and X/a, which is in this case allowable (see Intermediate Practical Physics, pp. 14 and 15). Hence the quantity which has to be added to the 42 MECHANICS AND GENERAL PHYSICS VII apparent mass W of the weights to obtain the true mass M of the body weighed is when (T represents the density of the weights, generally brass, and may be taken to be 8"4. The density \ will vary with the pressure and temperature of the air, and the amount of moisture present. It will be seen that the correction is positive or negative, according as the density of the substance weighed is less than or greater than that of brass. If the masses on one or both sides of the balance consist of different materials, the correction must be determined separately for each. Thus if the weights are partly of brass and partly of platinum, it may be necessary to take this into account, and the correction be- comes when Wi, o-^ are the mass and density of the brass weights, and TFs) o"2 those of the platinum weights. In the example given to illustrate Exercise II. of the present section, a mass of quartz weighing nearly 200 grms. is weighed to a tenth of a milligram, i.e., to about one part in two millions. The density of quartz being 2'65, the correction for buoyancy amounts to about 68 mgrms., and has therefore to be determined with an accuracy of one part in 680. This means that the temperature of the air in which the quartz is weighed, must be known to J of a degree, and its pressure to 1 millimetre of mercury. The amount of moisture present in the air should also be known, but no appreciable error will be committed if the air is assumed to be half saturated. It may also be verified that an error of one part in a thousand in the assumed densities of the quartz and the brass weights, would cause errors of '1 and '03 mgrm. respectively in the weighing. This shews how very difficult it is to weigh accurately to one part in a million. Before passing on to the Exercises in weighing, students should read carefully through the following instructions, which must be rigidly adhered to, as otherwise the balance may be seriously damaged. VII ACCURATE WEIGHING WITH THE BALANCE 43 Precautions necessary in weighing with a delicate balance. 1. Test whether the balance is in working condition, by lowering the arrestment carefully ; the pointer should slowly swing through a few divisions only. 2. See that the box of weights is complete, then place the riders on their supporting arms. 3. Do not touch the weights with your fingers but with the pincers or forks provided. 4. Xever place weights on the pans or take weights ojf, except when the balance is arrested. 5. The arrestment must be lowered with special care during the first stages of weighing, when the weights on the two sides are not yet nearly equal. Watch the pointer while the arrest- ment is lowered very slowly. As soon as the pointer is seen to start sharply to one side, raise the arrestment. Notice carefully whether the motion of the pointer to the left or right means that the weights placed in the pan are too small or too great. 6. If the arrestment is to be raised while the balance is swinging, wait till the pointer is nearly at its central position, then raise. The least possible injury will in this way be done to the knife edges. 7. The final weighings must be made with the balance case closed, and Ccire must be taken that the pans do not swing. Large swings of the pans should be carefully stopped by touching the pans with thumb and forefinger or with a camel hair brush while the beam is arrested. 8. In reading successive turning points take no account of the first, which is sometimes irregular. 9. When the weighing is complete, replace the weights carefully into their proper places in the box, remove the arrestment handle, place it in the balance case, and close the case. Exercise I. To weigh a body using the zero at no load. 1. Find the position of rest of the balance without load. Call this the zero of the balance at no load for the time being. 44 MECHANICS AND GENERAL PHYSICS VII 2. First ascertain by trial on a rough balance, that the weight of the given quartz crystal is say between 100 and 200 grms., then place it in one pan (the left for instance) and weigh to the nearest centigram in the following way : — Put 100 grms. in the right pan and add the weight which comes next in descending order of magnitude in the box of weights. Continue adding weights as you have been taught to do in the Intermediate Course (p. 50 and 51), until you find that the addition of another centigram shews excess of weight. Determine the weights on the pan, by noting the empty compartments in the box of weights, and record in your note-book. The number found must be checked by noting the weights themselves as they are removed from the pan at the end of the experiment. 3. Determine the weight to the nearest 2 milligrams. Use the rider for this purpose, placing it first at the point marked 6 on the beam, then at 8 or at 4, according as the additional 6 mgrms. have been found too small or too great. The student will have been able to proceed so far without taking readings of the pointer. But at this stage he will have to make a rough determination of the position of rest by taking the arithmetical mean of the readings of two consecutive turning points. As he gets nearer to the true value of the weight, he will have to determine the position of rest more accurately, and it will be necessary to read five turning points. Let it be found in this way, that the weight lies between 185-874 and 185-876 grms. 4. Determine the weight to the tenth part of a milligram. Observe accurately the positions of rest for the two weights differing by 2 milligrams between which the true value has just been found to lie. Then by interpolation calculate the weight which would bring the position of rest to the zero at no load. With very delicate balances it may be necessary to deter- mine the weight to the nearest milligram before proceeding to interpolation. 5. Again determine the zero at no load. Enter vour results as follows : — VII ACCURATE WEIGHING WITH THE BALANCE 45 5 Oct 1896. Balance: A. Zero of unloaded balance Position of rest with 185-878 grms. in right pan „ „ ,, » + 2 mgrms. „ „ 10-36 11-58 9-89 10-34 10-35 1-69 Zero of unloaded balance Mean zero at no load ... Difference produced by 2 mgrms. 11-58-9-89 = Additional weight required in scale divisions 1 1-58 - 10-35 = 1-23 2x1-23 „ m mgrms. ^ ^^ = ... I'O mgrs. Required weight ... ... ... ... 185-8795 grams. Exercise II. To weigh a body by the method of interchanges, sometimes called Gauss's method. The method given in Exercise I. does not correct for any of the errors of the balance, and if the weight is to be obtained accurately, it is necessary to adopt the method given in the present exercise, or one equivalent to it. The method, which has already been used in Exercise III. of the previous section, consists in interchanging the weights in the pans, and finding directly, or by interpolation, a weight which will bring the balance to the same position of rest, whether the substance to be weighed is in the right or in the left pan. It is clear that two weights which can be interchanged without altering the position of rest of the balance, must be equal. Proceed as follows : — 1. Find the weight to the neare.st 2 mgrms., as in Exer- cise I. 2. Find the position of rest when the substance is placed in the left pan, the lower one of the two limiting weights being placed on the right-hand side of the beam. 3. Interchange weights and substance and find the position of rest, being careful to remove the rider from the right-hand beam, and to place it or a .similar one at the corresponding point on the left-hand beam. 4. Interchange once more, so as to bring back the substance into the pan in which it was originally placed. 5. Increase the weight on the right side by 2 mgrms., and determine the position of rest. 46 MECHANICS AND GENERAL PHYSICS VII Reduce and enter as follows : 5 Oct. 1896. Balance: A. Position of rest with 185-878 grm. weights on right ... 10*57 (a) left ... 1000(6) „ „ right ... 10-45 (c) „ „ „ +2 mgrms. „ „ ... 8-94 Difference produced by 2 mgrms. ... ... .. 1-51 Mean of (a) and (6) 10-51 Zero of balance with load ... ... ... 10-26 Ji Additional weight required in scale divisions 10-26 - 1000 -26 2 X -26 _ „ „ „ „ mgrms. -p^ = ...-34 mgrms. Required weight ... ... ... ... ... 185-87834 grams. Note. — It may happen, if the arms of the balance are not sufficiently equal, that the weight does not in reality lie between the two limits found in Exercise I. If the real weight is above the higher limit, the additional weight required to produce the balance will be found greater than 2 mgrms. If, on the other hand, the true weight is smaller than the lower limit, the posi- tion of rest when the weights are on the right ((a) and (c) above) gives a lower reading than in (h) when the weights are in the left pan. Thus, suppose the reading for (6) had been H'O instead of lO'O we should have had to write : Additional weight required in scale divisions 10-26 — 11*00=— "74, the negative sign indicating that the correction has to be sub- tracted from 185*878. Students should note carefully whether the correction they find is to be added or subtracted. It will be observed that the substance is weighed twice in one pan and once in the other. The object of this is to eliminate the effects of changes in the balance which take place if the temperature of the balance case is increasing owing to the approach of the observer, or to the presence of gas flames. The difference between (a) and (c) in the above example, may either be due to accidental causes or to a systematic change. If the former, the mean will be a more probable value of the position of rest than either of the observed numbers ; if the latter, the mean will represent the position of rest at the time VII ACCURATE WEIGHING WITH THE BALANCE 47 at which the observation (6) was taken, provided the observa- tions were carried out at nearly equal intervals of time. In any case to get the best result the mean of the first and third observations should be compared with the second. If extreme accuracy is not required, the third observation may be dis- pensed with ; if, on the other hand, it is required to determine the position of rest correctly to the hundredth part of a scale division, it will be necessary to interchange the weights oftener. To complete the exercise, the result should be checked by placing the rider at the position corresponding to the weight found to the nearest tenth of a milligram (185*8783), and the weighing repeated. The barometer and thermometer should also be read, so that the buoyancy correction may be applied. The barometer need only be read once, unless there is reason to suppose that it is rapidly changing at the time, but the tem- perature of the balance case should be taken at the beginning and end of the experiment. An example will shew how the final result is now arrived at. 5 Oct., 1896. Balance: A. Assumed weight 185-8783 grs. Barometer 7 6 '4 cm. Time 11 h. 15 m. Temperature of balance case .. ... 18° -6 0. Position of rest with 185-8783 grs. on right 10-36 left 10-46 right 10-39 + 2 mgrms. „ 8-91 grs. , 10-41 11 h. 32 m. Temperature of balance case ... ... 18''-8 C. Mean position with weights on right (10*36 + 10-39)/2 = 10-375 left 10-46 Zero of loaded balance = (10375 + 10-46)/2 ... = 10-42 Difference produced by 2 mgrms. = 1040 - 8 91 = 1-49 Additional weight required in scale divisionsl 10-42-10-46 J ••• 2 X *04 Additional weight required in mgrms. . . - ... = '05 mgrms. Required weight 185*8783 - 00005 = 185*8783 grams. » >» »> » )) »> >> >> 48 MECHANICS AND GENERAL PHYSICS VII It will be seen that the repetition of the experiment has led to a result which is the same as that of the previous determina- tion, and if the experiments have been carried out with care, the difference between them should never exceed '2 mgrm. If the difference exceeds 25 mgrm. the rider should be placed at the position indicated by the last result, and a fresh determina- tion made. Buoyancy Correction. To calculate the buoyancy correction, first calculate by the following method the density of the air, assuming it to be half saturated. TABLE I. Density of dry air at ordinary temperatures and pressures. Temperature Pressure in cms. of mercury 73 cm. 74 cm. 75 cm. 76 cm. 77 cm. 10° c. 15 20 25 •001198 1177 1157 1138 1214 1193 1173 1153 1231 1209 1189 1169 1247 1226 1205 1184 1264 1242 1220 1200 TABLE IL Maximum pressure of water vapour at ordinary temperatures. Temperature Pressure =p 10° C. •91 cms. Hg. 12 1-04 „ 14 M9 „ 16 1-35 „ 18 1-53 „ 20 1-74 22 1-96 24 2-22 „ 26 2-50 VII ACCURATE WEIGHING WITH THE BALANCE 49 Consider a mixture of two gases the densities of which at a pressure P and given temperature, are d^ and rf^. Let the partial pressures of the two gases be p, and p^ respectively, the total pressure being P=^pi-hpi. The density X of the mixture is i.e., the density of the mixture is equal to that which the first constituent alone would have at the pressure If the two gases are air and aqueous vapour respectively, the index 2 referring to the latter, the ratio d^/d^ is very nearly 5/8, hence F + —j-^ p^^ P - ^p^. If then p^, the pressure of the aqueous vapour present in the atmosphere at the time, is known, we may use Table I. to determine the density X, by taking the air to be dry at a pressure 3 P— -pi, instead of saturated at the observed barometric pres- sure P, and if we assume the air to be half saturated, i.e., take P2 = ^, p being the maximum pressure possible at the observed temperature, given in Table II., we should have to take the 3 equivalent air pressure to be P — j^p. The value of p at the temperature of the room will on the average be about 15 mm.; the error we should make if the air happened to be totally dry or totally moist, would therefore be the same as if we had measured the height of the barometer incorrectly by about 3 mm., which would cause an error in the density of the air and in the buoyancy correction, of about one part in 250. If we require to make the correction with certainty to less than that amount, we should have to measure the pressure of aqueous vapour in the balance case. In the above example the barometer stood at 7638 cm., and the mean temperature of the balance case was 18°'7C. 8. p. 4 50 MECHANICS AND GENERAL PHYSICS VII Pressure of aqueous vapour at 18°7 C. ... .,, = 1'6 cm. (n)" = -3 P =76-38,, 3 P — Y^j9 (to the nearest millimetre) ... ... =76'1 „ Density of dry air at 18° C. and 76*1 cms. pressure (by interpolation) from Table I. ... ... ='001215 Density of dry air at 19" C. and 76*1 cms. pressure (by interpolation) from Table I. ... ... ="001211 Density of dry air at 18°*7 C. and 76*1 cms. pressure (by interpolation) from last two values ... ='001212 To find the buoyancy correction X ^ the specific gravity of the body weighed must be known approxi- mately. In the above example the body was Quartz, so that if W and p refer to the body weighed, Tfi and a^ to the brass weights, and TTg, a^ to the platinum weights, Z =185:88 p 2-653 Ii = l^ = 22-021 <"' 8'*^ [ = 22-06 I?=:!i = -04 <r, 21-5 ) Z_Z._I? =48-00 .-.buoyancy correction = '001212 x 48'00 = '05817 Weight found = 185'87825 Weight corrected (to the nearest tenth ofamgrm.) =185*9364 grams. It will be seen that if we had neglected the platinum weights, we should have made an error of about 1/20 mgrm., which, con- sidering the uncertainty in the assumed specific gravity of the VII ACCURATE WEIGHING WITH THE BALANCE 51 brass weights used, would have been allowable, but if we had assumed all the weights 185*88 to be brass, the error would have amounted to more than the tenth of a milligram. This example shews how difficult it is to obtain a weight correctly to one part in a million. Numerical Exercise, A litre of water is to be weighed to the nearest milligram ; calculate how nearly you require to know the height of the barometer, the temperature of the balance case, and the specific gravity of the brass weights. 4—2 SECTION VIII. DETERMINATION OF THE DENSITY OF A SOLID. Apparatus required : Delicate balance, piece of quartz, specific gravity flask, air and water hath^, small pieces of quartz. The density of a substance at any point, is defined to be the quotient of the mass of a small volume of the substance at that point, by the volume. If the substance is homogeneous and m is the total mass, and v the total volume, we have p = m/v. Hence if the unit of mass is the gram, and the unit of length the centimetre, the density of a homogeneous body will be numerically equal to the mass of one cubic centimetre of the substance. In the metric system, the gram was originally chosen to be the mass of 1 cubic centimetre of water at its point of maximum density 3°*95 C. A kilogram, equal to 1,000 grams, deposited in Paris, serves as the ultimate standard for weights constructed on the metric system. Since the experiments determining the gram were made, however, physical instru- ments and methods of observation have improved, so that a small difference is now found to exist between the theoretical gram and the practical standard of mass. In consequence, the density of water at 3°*95 C. is not unity, as it was meant to be, but is 1 '000013. The difference is so small that it may gene- rally be neglected, but it would have to be taken into account. VIII DETERMINATION OF THE DENSITY OF A SOLID 53 if for instance, 100 grams of water were to be weighed correctly to a milligram, and the volume occupied by the water calculated from the results to one part in 100,000. The specific gravity of a homogeneous substance, is defined to be the ratio of the mass of any volume of the sub- stance, to the mass of the same volume of water at 3°*95 C. As we may take the mass of 1 c.c. of water at 3°*95 C. to be unity, it is clear that the specific gravity is numerically equal to the density expressed in the C.G.s. system of units. This is an advantage, as in that system we may dispense altogether with the idea of "specific gravity" and always use that of " density " instead. Students must be clear, however, that the two terms are not synonymous, as the numerical value of the specific gravity is independent of the units of length and mass, while the number expressing the density will depend on those units. Since the direct determination of the volume of a body cannot be carried out accurately unless the body is of some regular shape, density determinations generally depend on a previous knowledge of the density of some standard substance, water being selected as the most convenient standard. Increase of temperature will in general diminish the density of a body, hence in stating the density, the temperature at which the number holds should always be specified. The density of water has been determined with great care, and has been found to decrease at an increasing rate per degree as the temperature rises. At 15° C, the ordinary temperature of the laboratory, the decrease of density of water for V C. is not much more than 15 parts in 100,000, while at 50° C. it is 6 parts in 1,000. Tables giving the densities at various tempera- tures are given by Volkmann {Wied. Aim. xiv. p. 206; 1881), and Landolt and Bornstein (Physikalisch-Chemische Tahellen), but the second column of Table 32 (Lupton's Tables) will be sufficient for most purposes. There are a number of different methods by means of which the density of a body may be determined, and the best manner of proceeding in each case will depend on the available quantity of the substance, on its chemical properties and state of aggre- 54 MECHANICS AND GENERAL PHYSICS VIII gation. We shall describe two methods, one of which will always be applicable if the substance is solid and insoluble in water. The method would have to be modified if the substance were soluble in water, or if it were hygroscopic. Method I. If a body is weighed first in air, and then suspended in a liquid, its apparent weight will be less in the second case than in the first, and it has been known since the time of Archimedes, that the apparent loss of weight is equal to the weight of the liquid displaced by the body. If M is the mass of the body, and p its density, its volume is M/p, and if (T is the density of the fluid in which it is weighed, the ap- parent decrease of mass will be M- , P The same holds for a weighing in air, the density \ of air being substituted for tr. Assuming the arms of the balance to be a and b cms. re- spectively, we have for the moments about the central knife- edge, of the forces on the two arms during the weighing in air, the quantities aMgll--") and hM,g(l-^y where ilf, is the apparent mass, and o-j the density of the weights. For equilibrium these moments must be equal, hence al/(l--)=61A(l-^ Similarly for the weighing in water, if ifg is the apparent mass Dividing the first equation by the difference between the first and second, we have a-\\ p) M^-M^' Or VIII DETERMINATION OF THE DENSITY OF A SOUD 55 i.e., the excess of density of body over that of air = - . — ^ X (excess of density of liquid over that of air). It will be noticed that neither the inequality of the arms of the balance, nor the buoyancy effect of the air on the brass weights, enters into the result, so long as the weights are always placed in the same pan. This is due to density determinations depending on ratios of weights only. It will also be seen that the equation giving p corrected for the buoyancy of the air, may be obtained from the equation in which the air is neglected, by subtracting the density of the air from each density occurring in the equation. This may be seen on consideration to be due to the fact, that the weight of a body obtained in air would be the same as the weight obtained in vacuo, if the density of the body as it was transferred from air to vacuo, were decreased by the density of the air. This holds for both the quartz and the water in the above case, and the principle will be used in other cases. The numerical calculation is best carried out by writing 1 — /c for <r, and p for MJ{M^ — M^ when we have p = p — p {k -^-X) -\-\ where the last two terms are small. Exercise I. Determination of the density of Quartz by weighing in water. 1. Take two pieces of fine silk thread 40 cms. long, use one to tie round the quartz crystal provided, leaving a length of about 16 cms. with a loop at the end, hanging from the crystal. Cut away all unnecessary thread, and cut off equal lengths from the other piece. 2. Suspend the quartz by means of the thread from the hook underneath the top of the support of the left-hand balance pan, place the other piece of thread in the right-hand jmn, and find the weights required to produce etiuilibrium. 56 MECHANICS AND GENERAL PHYSICS VIII 3. Remove the quartz from the balance, place it in water in a beaker, and boil the water to drive off the air bubbles adhering to the quartz. Then cool the water by pouring into it water from the tap gently without causing splashes. Place a small wooden stool across the left-band pan of the balance, so that the pan does not come into contact with it at any point. Support the beaker on this stool and suspend the quartz again from the hook. See that the quartz is entirely immersed in the water and that no bubbles of air adhere to it. Place a thermometer in the water, and note the temperature (Fig. 16). Cut off from the piece of thread in the right-hand pan a length equal to the length of thread in the water, and remove it. Weigh the quartz. Record as follows : — Density of Quartz. Method I. Date, Jan. 6th, 1893. Balance A. Box of Weights A. Weights in right-hand pan throughout. Temperature of Balance Case, 18°'4 C. Weight of Quartz in air (ifi) 40-882 grms. Temperature of Water, 20° C. Apparent Weight of Quartz in Water (Mg) ... 25*487 „ Loss of Weight = 15;395 „ M, 40-882 M^-M, 15-394 a = -9983 X=-0012 = 2-6556 o-_X = -997l M ^^^(.-X) = 2-6479 X= -0012 p = 2 6491 Hence p the density of the quartz at 20° C. = 2*649 1. VIII DETERMINATION OF THE DENSITY OF A SOLID 57 Method II. If the solid can only be obtained in small pieces, we may determine the density by the use of a "specific gravity flask " (Fig. 19), which is a small glass flask provided with a well-ground stopper traversed by a narrow channel. When it is filled with a liquid, and the stopper is inserted care- fully so as to exclude air bubbles, the excess of liquid will flow out through the capillary open- ing. By means of a piece of blotting-paper a small quantity of the liquid may be removed, so that it just reaches to a marked height in the opening. The flask may in this way be repeatedly filled to the same level, and if its temperature is the same, the volume of its contents will be the same. The flask having been cleaned and dried, the density re- quired is determined by the following series of weighings, the letters, F, &c., representing the weights obtained : — 1. The flask dry, F. 2. The flask dry with the dry solid placed inside, F-\- M^. 3. The flask with the solid inside, after filling up to the mark with a liquid of known density a^ at a tempera- ture ^, F-\-M^-\- W^. 4. The flask entirely filled up to the mark with a liquid of density Cj at a temperature f.^, F ■\- W^. Since a^ is the density of the liquid at fj, the volume of the W flask at ^3, neglecting the effect of the air on the weighing, is — - , and the volume at ti will be — ^ ^ - ^ , where a is the coefl^cient <r, 1 + orfg of cubical expansion of glass. Similarly the volume occupied by the liquid at <i = . The difference between these volumes is the volume occupied by the solid at ^, and the density p of the solid at t, is the mass divided by this volume. The effect of the buoyancy of the air may be taken into 58 MECHANICS AND GENERAL PHYSICS VIII account by subtracting the density of air X, from each of the densities in the equation for p (page 55), and we thus get P-^= W„. l + at. tV. • cTo — X ' 1 4- olL o-j — \ If we take weighings of the flask empty and when filled with each of two liquids, we may compare the densities of the liquids, since on making M^ = in the above equation, we have W^ 1 + a^ ^ Tf 1 0-2 — X ' 1 4- a^o cTi — \ ' an equation from which we can determine one density if the other is known. If in the former equation, we take the liquid to be water in each case, so that cr^ and a^ are both nearly unity, we have, using the methods of approximation given in Intermediate Practical Physics (page 16), P — \= == ^ . ( <Ji — A, ). ^ W,(l + a;-a,-hOit,-t,)-W, Or writing o-j = 1 — /c, and ^ W^(l-\-<r^-(r^-{-at,-t^)-W^' we have p = p - p (/Ci + X) + X. Exercise II. Determination of the density of Quartz by the specific gravity flask method. 1. Clean the 50 gram flask (Fig. 19) provided, by washing it if necessary with a strong solution of caustic potash, and then with water from the tap. The potash is to be thoroughly removed with tap water, and the final washing made with dis- tilled water. Place the flask on the shelf of an air bath kept at about 120° C, insert a glass tube into the flask, through an opening in the top of the bath. Look at the flask occasionally, and when the drops of water have evaporated from the sides, draw the moist air out of the flask, by applying the mouth to the VIII DETERMINATION OF THE DENSITY OF A SOLID 59 Upper end of the glass tube. Repeat this several times, then remove the flask and allow it to cool. If no moisture is de- posited on its inside surface as it cools, place it in the left-hand pan of the balance, and weigh. With the quantities of the substance available, it will be suflScient to weigh accurately to the nearest milligram, and the process of weighing may therefore be shortened by taking only three readings of the vibrating pointer (two on one side and one on the other), instead of five, as in the previous exercises. 2. Dry thoroughly 15 or 20 grams of the broken up pieces of quartz with which you are provided, and place in the flask. Weigh the flask and contents, and hence deduce the weight of the quartz. 3. Pour some distilled water into the flask. If air bubbles adhere to the small pieces of quartz, it will be necessary to expel them. For this purpose place the flask, with the solid pieces completely covered with water, in the air bath, and heat up carefully until the water just begins to boil. Take out the flask, and after allowing a little time for cooling, fill up with distilled water, and then place it in the water bath with which you are provided, keeping it in position by indiarubber bands. Stir the water well, and keep it at a temperature two or three degrees higher than that of the room. Read the temperature to 0°05 C. The temperature of the bath must be kept constant, and the flask kept in the bath for a sufficient length of time to allow the water in it to take up the same temperature. This may be ascertained by placing a thermometer in the flask. The stopper is then inserted, and the quantity of water in the flask is adjusted so that it reaches to the mark across the capillary opening in the stopper. Take the flask out of the water and dry the outside ctxrefully, taking care to avoid heating it by contact with the hand and forcing out any of the water. The object of filling the flask with water at a temperature above that of the air is now apparent, for when the flask is taken out of the water bath, the water in it will contract, and thus the danger of losing any 60 MECHANICS AND GENERAL PHYSICS VIII by accidental heating of the flask is greatly diminished. After the flask has been dried on the outside, leave it in the balance case for a few minutes so that it may acquire the temperature of the balance case, then weigh again. If the flask is not perfectly dry outside, evaporation will take place and the Aveight of the flask will slowly diminish. Ascertain that the weight remains constant. The difference between the weights of the flask in this and the previous weighing, will give the weight W^ of the water it contains. 4. Now take out the quartz, fill the flask with distilled water and place it in the water bath. Keep the temperature of the bath nearly the same as it was in the second weighing ; a correction will be necessary if any difference exists between the two temperatures, which should be known to the twentieth of a degree. Close the flask, dry and weigh it. Record as follows : — Density of Quartz. Method II. Date, Jan. 6th, 1893. Balance A. Weights A. Flask 13. Weights in right-hand pan throughout. Temperature of Balance Case, 16"'0 C. Weight of flask, i^ Weight of flask and quartz, F-\-Mi Hence weight of quartz, M^ Weight of flask, quartz, and water) Sitt,= lS°,F-\-M,+ Wr J Hence weight of water at 18°, Wi Weight of flask filled with water [ 8itt,=^19°\F+W^ J Hence weight of water filling flask ) at 19°, W, J 17-325 grms. 32012 „ 14-687 = 76-438 = 44-426 67-274 49-949 1 This temperature differs more from t^ than it need be allowed to do. The difiference is taken great here to shew clearly the magnitude of the correction introduced. VIII DETERMINATION OF THE DENSITY OF A SOLID 61 At 18^ <T, = -99867 At 19°, Co = -99847 o-i-o-., = -00020 a =000023 a{t,-Q = -00002 Sum = 00022 IT, (o-j — tTa + a^ — tj) = Oil grms. W^il-^a.-a. + aU-U) = 49-960 Fi = 44426 Tr,(l + <r,-o-2 + a^a-^)-Trx = 5-534 "-■^ - ^ K, = -00133 X = -00122 K^ + \ = -00255 /)i(/tfi + X) = 0067 X = 0012 Hence density p of the quartz at 18° C. = 2*6485 SECTION IX. DETERMINATION OF THE DENSITY OF A LIQUID. Apparatus required : Balance, specific gravity flask, salt solution, water bath, Mohr's balance, and hydrometer. Method I. By the specific gravity flask. In the previous section (p. 58) it has been shewn how the density of a liquid may be found by means of the specific gravity flask, and we now proceed to apply the method to the determination of the density of the salt solution provided. Make use of the specific gravity flask used in the deter- mination of the density of quartz. The apparent weight of the flask empty was found to be F grams, and the apparent weight of water filling it at the temperature ^i, TTj grams. The flask should be dried, filled with salt solution, and placed in the water bath at a temperature 2 or 3 degrees higher than that of the balance case. After allowing the solution to take up the temperature of the bath, put in the stopper and carefully remove with filter paper the drop at the top of the hole in the stopper. Let the observed temperature of the water bath be fg- Remove the flask, dry the outside and weigh. Let TTg be the apparent weight of the salt solution. W The volume of the flask at the temperature ^i is — ^ when <7i is the density of water at that temperature ; the volume of W the flask at 0° C. will therefore be — ^ (1 + a^i), where a is the IX DETERMINATION OF THE DENSITY OF A LIQUID 63 coefficient of expansion of glass, and this volume must be the same as that calculated from the corresponding weighing of the liquid, i.e. - " (1 + a^). The correction of the weighings for buoyancy is introduced by writing o-, — \ for a^ and a^ — X for o-j (see page 55), hence (Ta — X 1 + af 1 * cTi — \ ' Or since a^i, aL are both small which equation serves to calculate a^ if o-j is known. Record as follows : — Density of Salt Solution. Method I. Date, Jan. 8th, 1893. Balance A. Weights A. Flask 13. Weights in right-hand pan throughout. Temperature of Balance Case, 18° C. Weight of dry flask, F = 17-325 grams „ flask filled with water at 19° = 67-274 .. „ water filling flask at 19° = TTj = 49949 „ flask filled with salt solution at 19°-2 = 70-282 „ salt solution filling flask at 19°-2 = F, = 52*957 Density of water at 19° = a, = '9985 „ of air at 18° =\ = 0012 .'. <7,-\ ... = -9973 a for glass = 000023, .-. o^ - <2= 0. tf;'4T949 = ^^^^^ <r,-X= 1-0601 X -9973 ... = 10572 X = 0012 .•.o-a=!density of solution ... = 10584 64 MECHANICS AND GENERAL PHYSICS IX Fig. 20. Method II. By Mohr's Balance. Mohr's balance (Fig. 20) is an ordinary balance modified so as to enable determinations of densities of liquids to be made rapidly. One arm of the balance is divided into 10 equal parts, and carries, suspended from its end by a fine silk fibre, a glass thermometer which is immersed in and indicates the temperature of the liquid. The other arm carries a counterpoise, the end of which is pointed, and comes close to a corresponding pointer on the frame of the balance. The balance can be clamped at any height by means of the screw in the stem. Place the balance on the stand in such a way that the levelling screw in the stand lies in the vertical plane through the beam. Suspend the thermometer from the graduated arm, and turn the levelling screw till the pointers are in line with each other. Fill the small test tube with water at 15° C. Raise the balance by means of the expanding stem, place the test tube under the thermometer, and lower the balance till the thermometer is entirely immersed in the water. The balance will no longer be in equilibrium owing to the upward force of the water on the thermometer, which we have seen is equal to the weight of the water displaced by the thermometer. To produce equilibrium weights must be placed in the notches of the arm of the balance. The largest brass weights provided are equal, the other weights are -^ and j^ respectively of the largest weights. The notches are marked from the centre of the arm to the end 1, 2, 3 9, the hook under the end of the arm being at the tenth notch. If we call the weight of the largest brass pieces 1, then if 1 is placed in the notch 6 it is equivalent to a weight of '6 placed at the end, and so on. Thus the weight IX DETERMINATION OF THE DENSITY OF A LIQUID 65 necessary to produce equilibrium is given by the readings of the notches in which the weights in order of magnitude are placed, the unit in which the weight is expressed being that of the largest brass weight. The weight found is in terms of this unit that of a volume of water at 15° equal to the volume of the thermometer. Remove the weights, raise the balance, and after drying the test tube and the thermometer replace the water by the liquid the density of which is required and see that the thermometer in the liquid indicates again 15° C. Again determine the weight to produce equilibrium. Since this is the weight of a volume of the liquid equal to that of the thermometer, the ratio of the density of the liquid at 15° to that of water at 15° is the ratio of the two weights. The density of the liquid at 15° is therefore the product of this ratio and the density of water at 15° C. (= -999). The weights are generally arranged so that one of the heaviest will when hung in the hook produce equilibrium when the thermometer is in water. In this case the density of the liquid can, to within 001, be read off immediately from the positions of the weights when the thermometer is in the liquid. Method III. By the Hydrometer. When a solid floats on a liquid it is acted on by two forces, one its weight downwards, and the other the pressure of the liquid upwards. Since the body is in equilibrium these two forces must be equal. But the pressure of the liquid is equal to the weight of the liquid displaced. Hence if W be the mass of the solid, V the volume of liquid displaced, <r its density, we must have W = Va, or (7 Hence when the solid floats on a denser liquid it sinks less than it does in a lighter liquid and the volume immersed varies inversely as the density of the liquid. 8. p. 6 66 MECHANICS AND GENERAL PHYSICS IX The Hydrometer provided (Fig. 21) has been graduated by being placed in liquids the densities of which had been determined by the previous methods. Hence to get the density of a liquid it is simply necessary to place the hydrometer in it and read the position of the surface of the liquid on the graduated stem. A test tube mounted on a block of wood is provided for holding the liquid. To read the scale, hold the eye below the level of the surface of the liquid, and gradu- ally raise it till that surface is seen foreshortened into a J\ straight line. The position of this line on the scale is the density required. Collect your results as follows : — Density of salt solution by flask ... ... 1'0584 \y „ „ „ Mohr's balance ... 1'059 ^^ hydrometer ... 1-058 Fig. 21. SECTION X. MOMENTS OF INERTIA. The following pages contain a short statement of the prin- cipal propositions concerning Moments of Inertia and the Compound Pendulum. Students should read them carefully and work out the examples before proceeding to the practical exercises on Moments of Inertia and the Pendulum. Some of the propositions are given without proof, and the students are referred to books on Dynamics {e.g. Worthington, Dynamics of Rotation ; Hicks, Elementary Dynamics) for a more detailed treatment. The Moment of Inertia of a particle about an axis is defined as 7nr^, where m is the mass of a particle and r the perpendicular distance between the particle and the axis. The Moment of Inertia of a number of particles about an axis is the sum of the Moment of Inertia of each. Thus particles 7/1,, m,, m,, at distances r^^r^, ?•,, r^, from an axis have a moment of inertia about that axis equal to niirx^ + nu^r^* + w,r,' + . . . . From this it follows that the moment of inertia of a body about any axis is equal to the sum of the moments of inertia of the separate parts of the body about the same axis, it can there- fore be found by dividing the body into a number of small parts and adding the products of the masses of the parts into the squares of their perpendicular distances from the axis. 5—2 68 MECHANICS AND GENERAL PHYSICS X The following moments of inertia are frequently required: — M always denotes the mass of the body which is supposed to be of uniform density. I. A right solid parallelepiped, whose edges are 2a, 26, 2c, about an axis through its centre m^^'^^^ perpendicular to the plane containing the edges 3 b and c. II. A solid cylinder of radius r about its ^r* axis of figure. 2 III. A solid cylinder of length 2a and radius r about an axis through its centre perpendicular M ( — -^'—j to the length of the cylinder. IV. A sphere of radius a about any axis ^ 2a^ through its centre. 5 The radius of gyration of a body with respect to an axis is the distance at which a particle of the same mass as the body would have to be placed in order to have the same moment of inertia. Hence, if the Moment of Inertia is written in the form Mk^, M being the mass of the body, k will be the radius of gyration. The moment of inertia of a body about any axis is equal to the moment of inertia about a parallel axis through its centre of mass, together with the product of the mass of the body and the square of the distance between the axes. The theorem of p. 67 enables us to calculate the moment of inertia of bodies from which parts have been removed, if the moments of the complete bodies and of the removed parts are known separately. Thus let it be required to find the moment of inertia of a circular ring about an axis passing through its centre and per- pendicular to the plane of the ring. Let the inner and outer radii of the ring be r^ and r^, and imagine the inner space to be filled with matter of the same density so as to convert the ring into a disc. Let rrii be the X MOMENTS OF INERTIA 69 mass of the smaller disc of radius 7\, which fits into the circular hollow of the ring, and let m^ be the mass of the complete disc of radius r,. Then from the proposition, if / is the unknown moment of the ring / + __== ^i,^^, or /= 2 • But if p is the density of the ring T/ii = irpr-^ and m.^ = irpr^. Hence If 3/ is the mass of the ring M = irp {r^ — r^% hence finally Exetxises. 1. Find the moment of inertia of a solid parallelepiped about an edge. 2. Find the moment of inertia of a cube with a central spherical hollow about an edge of the cube. 3. The moment of inertia of a hollow circular cylinder is required about an axis at right angles to the length of the cylinder and passing through one of its end planes. The moments of inertia of solid bodies are of importance as these quantities often occur in physical problems. Thus the Kinetic Energy of a body rotating about an axis is expressed by \Iw*, when / is the moment of inertia of the body about the axis, and w is the angular velocity. If a body oscillates about an axis under the influence of ex- ternal forces, as for example a pendulum under the influence of gravity, the time of vibration depends on the moment of inertia. Oscillations like those of a pendulum occur whenever a body is capable of rotating about an axis, and is acted on by forces \ 70 BfECHANICS AND GENERAL PHYSICS X such that if the body is disturbed from its position of equili- brium, a couple acts on the body tending to bring it back. If the couple is proportional to the angle through which the body is turned, the oscillations are strictly isochronous, that is, the time of vibration is independent of the amplitude of oscillation. This is only approximately true in the case of the pendulum, but much more nearly true if the body is capable of oscillating in a horizontal plane, being brought back to its position of equilibrium by the torsional forces of a wire by which the body is suspended. The time of vibration is then given by the equation 3^ = 2^5^' where / is the moment of inertia of the body about the axis around which it turns, and N is the couple per unit angle of deflection, tending to bring the body back. SECTION XI. EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA. Apparatus required : Rectangular metal block, fine loire, and hi filar suppoi'ts. The moment of inertia of a body about an axis through its centre of gravity, may be found experimentally by suspending the body, either by a single fibre or by a double one, in such a way that it can perform torsional oscillations about that axis, and determining the time of an oscillation. A second body, the moment of inertia of which, about some axis through its centre of gravity is known, either on account of its regular shape or by previous determination, is then attached to the first, in such a way that the axis is coincident with that of the fibre, and the time of an oscillation again determined. From these times the moment of inertia of the first body can be found in terras of that of the second by the equation : /„ = I-^Tq^JT-^ — Tq^, where Ii is the moment of inertia of the mass added, To and Ti are the times of oscillation. This is the method often used when the moment of inertia of a suspended magnet is to be determined. If the bifilar method of suspension is adopted, the body, the moment of which is required, is laid in a stirrup at the lower end of the suspension, with the axis about which the moment is required vertical and half-way between the two suspending fibres. If it is rotated in a horizontal plane through a small angle, the suspension resists the rotation with a force propor- tional to the sine of the angle of twist. The couple N for small angles of rotation being = Mg ~j ' , we obtain the time of vibration by substituting this value in the general equation where M = mass of body suspended. 72 MECHANICS AND GENERAL PHYSICS XI I = moment of inertia of body suspended about axis of suspension. I = length of each fibre. di,di = distances apart at top and bottom. g = gravitational acceleration. Since / = Mk^ where k is the radius of gyration of the body about the axis we have for a small oscillation V gdido' Exercise I. To verify the relation between the time of oscillation and the constants of the bifilar suspension. Suspend from the lower end of the fibres the rectangular block provided (Fig. 22), passing the fibre through the holes at the ends of the small brass strip which can be screwed to the block, then over the larger pulley above the separated horizontal clamps forming the upper support. When the block has come to rest, screw up the clamps so that the two fibres are held firmly, then set the block in oscillation through an arc of about 20°, and determine the time of oscillation and the mean arc of twist on each side of the equilibrium position (Section III). Measure the length of the fibres and their distances apart at the top and bottom. Now unscrew the clamp, pass the fibre through the holes nearer the centre of the strip, and over the smaller pulley, then clamp it, measure the length of the fibres and the distances apart at top and bottom, and again determine the time of vibration. Reduce the length of the fibres by moviog the clamp to a lower part of the support and again clamping the fibres. Again determine the time, and measure the distances apart of the fibres. Fig. 22. lengths and XI EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA 73 Tabulate the observations and results as follows :- 3 Jan., 1898. Apparatus A. Length cms. Distances apart cms. did. ^ Axis of block vertical Time seconds (time)5 ^. Top Bottom 76-0 75-7 41-5 3-48 1-50 1-50 3-53 1-40 1-40 12-28 -161 2-10 -0277 2-10 ; -0506 longest 2-927 6-960 5-201 8-57 48-35 27-00 1-38 1-34 1-37 The constancy of the numbers in the last column verifies the law for the bifilar suspension. Exercise II. To determine a Moment of Inertia. The rectangular metal block provided may be attached to the bifilar suspension, so that its three principal axes coincide in turn with the vertical axis of the suspension. The previous observations suffice to determine the moment of inertia about its longest axis. Attach the block to the suspension so that the axis of mean length is vertical, and determine the time of oscillation, using the full length of fibre and maximum distance apart. Next attach the block with its shortest axis vertical and determine the time. Calculate from the observation the moments about the three principal axes, arranging the work as follows : — Block marked A. 3 Jan., 1898. I = 760 cms. d, = 3-48) rf, = 3-53 did, I = 161, = 40-2. Axis vertical Time see. f« At* 4ir« Al I observed I calculated long mean short 2-927 4-33 5-128 8-57 18-49 26-30 •21 •46 •65 8-6 18-6 •26-6 1026 1010 2076 2056 3120 3182 The fifth column gives the values of K 74 MECHANICS AND GENERAL PHYSICS XI Weigh the metal block, multiply each value of k- by the mass to get the moments of inertia, and tabulate as in the sixth column above. Take the dimensions of the block, and calculate the moments by the formula where a, 6, c are the half lengths of the sides of the block. Record as follows and enter in preceding table. jlf= 1200 grams a = 7-5 cms. .'. a^ = bQ-^ .\ ia = 1010 6 = 5-5 62 = 3Q.2 /j, = 2056 c = -55 c2= -3 /o = :3182 The oscillations of a bifilar suspension are like those of a pendulum only approximately isochronous, and the arc of rotation should therefore be small if great accuracy is required. If the arcs are large the correction given in the next exercise may be applied. SECTION XII. THE COMPOUND PENDULUM. A Compound Pendulum is one in which the mass is distri- buted over a finite volume and not concentrated at a single point as in the simple pendulum. Strictly speaking every pendulum is a compound pendulum, and we can only approach the ideal simple pendulum by reducing the volume of the heavy bob of an actual pendulum as much as possible. In Fig. 23 let G be the centre of mass of a heavy body of mass in capable of turning about an axis through at right angles to the plane of the paper. In the position of equilibrium, the centre of mass G will be vertically under 0. If the body is displaced through an angle 6, the resultant gravitational force passing through G, will have a moment mgh sin 6 about 0, g being the ac- celeration of g^vity and h the distance OG between the centre of mass and the axis of rotation. This couple is proportional to sin 6, which if 6 is small will be nearly equal to 6 in circular measure, and the couple tending to bring the body back into its position of rest, will be nearly mghS, i.e. nearly proportional to the angle of displacement. Under these circumstances the body will o.scillate about the point 0, and the time of oscillation will for small displacements be independent of the amount of the displacement, i.e. the oscillations will be isochronous. Fig. 28. 76 MECHANICS AND GENERAL PHYSICS XII The time of oscillation will be V mgh' where / is the moment of inertia of the pendulum about the axis of oscillation, and mgh the couple for a displacement of a right angle. If K is the radius of gyration of the body about 0, I = mK^ and the equation becomes Definition. The length of the equivalent simple pendulum is the length of the simple pendulum having the same time of oscillation as the given body. As a simple pendulum of length I has a time of oscillation equal to 27r . / - , it follows that if I be the length of the equi- V if valent simple pendulum I = K'lh. If m^'2 be the moment of inertia of the compound pendulum about an axis parallel to the axis of oscillation, and passing through the centre of mass G, we have K^^k' + hK Hence i.e. (I -h)h = k\ If a point P be taken in the line 00, Fig. 23, such that OP is equal to the length I of the equivalent simple pendulum, P is called the centre of oscillation, while is called the centre of suspension. Hence the radius of gyration with respect to the centre of mass, is the geometrical mean between the distances of the centre of mass from the centres of suspension and oscillation respectively. If the body is suspended from P, will become the centre of oscillation, for k^ being a constant, h and l — h may be inter- changed without interfering with the truth of the equation. XII THE COMPOUND PENDULUM 77 Let a pendulum be constructed, Fig. 24, such that it is capable of vibrating about either of two knife-edges A and B, the line joining the edges passing through G the centre of gravity of the pendulum. If the knife-edges can be adjusted so that the time of oscil- lation is the same whether the body oscillate about A or about B, then unless A and B are equidistant from G, the length AB is that of the equivalent simple pendulum, and as that distance can be accurately measured the value of g can be found accurately from the equation Fig. 24. This is the principle of Kater's Pendulum. The isochronism of the pendulum has been stated to depend on the equality of sin Q and 6 for small values of 0. If Q be- comes large the quantities are no longer equal, and the time of oscillation will vary with the angle of displacement. In that case a closer approximation is given by ^-^Vk.^S). where Q is the semi-arc of oscillation. Hence if T,^ is the time of an infinitely small oscillation and T the observed time we may write ^0 = ^(1 — ^^). The following table gives the value of ^/16 for different arcs of swing. 6 ^/16 r 10- 20' •000019 •000476 •001904 •007615 In general there are four possible centres of oscillation on 78 MECHANICS AND GENERAL PHYSICS XII every straight line, such that the times of oscillation of the body about them are equal. Let A be any point of the body, G the centre of mass, and let the axis of rotation be at right angles to the plane of the paper (Fig. 25). With G as centre, draw a circle through A, join AG and produce to cut the circle again in A' . Then since GA' = GA the time of oscillation about A' will be the same as that about A. If AG = h and k is the radius of gyration about the axis passing through G, the length of the equivalent pendu- lum will be {k^-{-h^)/h. Make AB equal to this length. Then if the centre of the suspension is at B, the time of oscillation is the same as before, and therefore the same also for any point B' on the circle passing through B and having G as centre. Two circles may therefore be drawn such that the times of oscillation about all points of them are the same. A straight line such as CG\ not necessarily passing through G, will inter- sect these circles in four points, about which the times of oscilla- tion will be equal. Exercise. Calculate the length of the equivalent simple pendulum of: — 1. A brass rod about its ends. 2. A brass rod about a point half-way between its ends and its centre. 3. A sphere about a tangent. 4. A cube about one of its edges. 5. A cylinder about one of its generating lines. SECTION XIII. EXPERIMENTAL DETERMINATION OF THE EQUIVALENT SIMPLE PENDULUM. Apparatus required : Brass bar with sliding knife-edges , support, and simple pendulum. A brass bar, in one end of which a number of holes have been drilled in order to make it unsymmetrical, is provided (Fig. 26). On it slides a knife-edge, so that it can be set swinging about an axis at right angles to its length at a distance from its centre of gravity which can be varied. In front of this brass bar a leaden ball is sus- pended by means of a thread, the length of which is adjustable. The experiment consists in adjusting the length of this simple pendulum till it has the same time of oscillation as the compound pendulum. Place the knife-edge about a cm. from one end of the bar and suspend the bar in its support. Make the thread of the simple pendulum about I the length of the bar, push both pendulums aside with the palm of one hand, and suddenly withdraw the hand so that the pendulums start simul- taneously. Watch the oscillations of both pendu- lums, noting which of the two gains on the other. If after a few oscillations one is decidedly ahead of the other, the simple pendulum must be length- ened if it swings too quickly, and shortened if it '** swings' too slowly. After a few trials the length may be adjusted so as to make the times of oscillation of the two pendulums equal to each other to within 2 or 3 per cent. When that is the case, set the pendulums swinging together once more, count the oscillations of the bar, noting again 80 MECHANICS AND GENERAL PHYSICS XIII which pendulum gains on the other; go on counting the oscillations until the pendulums swing again together. If n is the number of oscillations of the compound pendulum counted, the simple pendulum will have performed (?i — 1) or (7i-f 1) oscillations, according as it is too long or too short. The number n should not be less than 30, otherwise a readjust- ment of the simple pendulum should be made. If one pendulum gains so slowly on the other that the number n cannot be fixed to one or two units, find ?ii and n^, so that i^ is the number at which it begins to be doubtful whether the pendulums swing together, and iu the number at which it begins to be certain that they have ceased to swing together; n will then be very nearly equal to (?ii + n^l% If V is the actual length of the simple pendulum, the length I of the simple pendulum which would swing exactly in the same time as the brass bar is equal to T f 1 j or T ( 1 + J , according as the simple pendulum made ?i — 1 or ?2 + 1 oscilla- tions, while the compound pendulum made n. If k is the radius of gyration of the bar about its centre of mass (r, and h the distance of the centre of mass from the point of suspension 0, then, as shown in the previous section, PO x OG = k^, which may therefore be determined from the observations made. Measure the distance of the knife-edge from the top of the knife-edge clamp, and the distance from the top of the clamp to the upper end of the bar. The sum of these is the distance -4 of the knife-edge from the top of the bar. Five positions of the knife-edge should be taken, about 1, 10, 20, 30, and 35 cms. from the top of the bar ; the bar must then be reversed and five ^^8- ^7. more observations taken. Determine the centre of mass G of the bar by removing the knife-edges and balancing the bar on one of them. Calculate for each experiment the distance OG. From this and the measurement of I calculate PG. XIII EQUIVALENT SIMPLE PENDULUM Enter as follows : — 21 Jan., 1894. Bar a. Distance of centre of mass from solid end of bar = 50*8 cms. „ from top of clamp to knife-edge ... = '9 „ Solid End of Bar at Top. 81 Distance top of bar to top of ! clamp Distance from top to knife- AG-AO = 0O n 1 Simple ' pendulum 1 obeerred I I' 1 1 n-l or n-fl l=OP OP-OO =P0 00.P0 cm. cm. cm. cm. 1 cm. cm. •7 1-6 49-2 38 63-8 39 67-2 180 885 8-2 9-1 41-7 36 59-5 1 37 62-8 2M 879 18-6 19-5 31-3 33 55-8 34 59-2 27-9 873 29-8 30-7 201 39 60-2 40 63-3 43-2 868 34-4 35-3 15-5 33 6G-5 , 34 70-6 55-1 854 Mean 872 Draw up a similar Table of results when the solid end is at the bottom. Draw a curve to express the above results, taking distances of the point of suspension from one end of the bar as abscissae, and the lengths of the equivalent simple pendulums for the points as ordinates, as shewn below : — = T == =EE::;:::::::::: ::::: ■± :::::: — :::± .:_. - "- :± _., :1 _-- ::::::::::2:::: ;::::::::;!::: : .i^::-_,z . ^. . _ ^! 9T »• "* 1. ^ . ,- "^ M • ::!::::::""::::::"":::"::::: ::8::::3£:::i E2ii:2E:::] a:::: :::J::::::::ffi:::: :l :: 1 tit 1 Fig. 28. 8. P. SECTION XIV. DETERMINATION OF YOUNG'S MODULUS BY THE BENDING OF BEAMS. Apparatus required : Uniform wooden beam and supports, weights, silvered glass scale and support. When a beam of breadth b and height h, supported at two points I apart, carries a mass ^ at a point half-way between the supports, this point is depressed by an amount d given by the equation where e is Young's modulus for the material of the beam, and g the gravitational acceleration. To verify this relation, place the two supports provided, so that the top edges are in the same horizontal plane and about 80 cms. apart. Look across them to see that they are parallel. Place the wooden beam so that the marks near its ends come over the supports, and behind its middle point place a graduated mirror with the scale vertical (Fig. 29). Arrange the rod so that its greater breadth is horizontal, and read on the scale the height of its middle point. By means of a hook suspend a 500 gram weight from the middle of the beam, and read on the scale the depression of the beam. Increase the weight to 1000 grams, and again read the position of the middle point. Verify, by means of your obser- vations, that the deflection is proportional to the deflecting weight, taking into account the weight of the hook. XIV YOUNGS MODULUS BY BENDING 83 Turn the beam so that its greater breadth is vertical, and repeat the observations. ^ Fig. 29. Use a shorter length of the beam, and with the greater breadth horizontal repeat the observations. From the observations calculate, using the above formula, Young s modulus for the material of the rod. Arrange your results as follows : — 3 Feb., 1899. Beam of Oak Wood marked 0. 3. Length of beam between supports ... =79*8 cms. Greater breadth ... ... ...= 1*36 cms. Lesser breadth ... ... ... = '88 cm. Greater breadth horizontal. Load in grams Reading at centre Depression cms. Depression per gram 7-90 509 6-65 1-25 •00246 7-90 963 5-55 2-45 •00254 7-90 Mean •00250 6—2 84 MECHANICS AND GENERAL PHYSICS XIV 1 981 (79-8)3 4 '0025 1-36 X (•88)' = 5'38 X 10*" dynes per sq. cm. Similarly for the observations taken with the greater breadth vertical and with the shorter length of beam, the values of e being collected as follows: — Oak Beam, 0-3. e in dynes per sq. cm. Greater breadth, horizontal „ „ vertical Shorter beam 5-38 X lO'** 5-24 5-29 Experiments with the shortened length of beam are here introduced in order to illustrate the law according to which the deflection is proportional to the cube of the length. If the sole purpose of the exercise had been to determine Young's modulus, these experiments would not have been necessary. Place the beam in its first position, and after observing the reading on the scale, place the 500 grams weight at a point which divides the beam in the ratio 1 : 3. Read the deflection at the middle point. Increase the weight to 1000 grams, and again read the deflection. Now place the mirror behind the point at which the weight has been applied, and remove the weight to the middle. Read the deflections for 500 and for 1000 grams, enter as below and Load Beading Deflection cms. Oat A 500 gr. at A lOOOgr. at A 7-85 at B 6-70 at B 5-65 at B 115 at B 2-20 at B OatB 500 gr. at B 1000 gr. at B 7-55 at A 6-40 at A 5-40 at A M5at A 2-15 at A XIV young's modulus by bending 85 verify that they are the same as those found in the former case. Hence the deflections at B due to a weight at A is equal to the deflection at A due to the same weight at B. If metal bars are used the deflections will be smaller than in the case of wood, and must be read off by means of some form of reading microscope. Young's modulus may also be measured if the beam is fixed at one end and a weight is attached to the other end, the relation between Young's modulus and the deflection at the point of application of the force is in that case given by d = 4!W.g -rrn • SECTION XV. MODULUS OF RIGIDITY. Apparatus required: Uniform vnreSy cylindrical weight, and watch with seconds hand. The elastic reactions of a homogeneous body subject to strain depend on two constants, the bulk modulus, or resistance to change of volume, and the modulus of rigidit}^, or resistance to change of shape. If a cylinder AB oi radius r and length I has one of its ends A fixed, while the other B is twisted, the angle of twist (f) is connected with the moment of the twisting couple P, and the modulus of rigidity n by the relation Hence if P can be measured, n can be calculated from the dimensions of the cylinder. If the cylinder is thin, e.g. a wire, P may be determined by suspend- ing a body the moment of inertia of which is known from the wire, and letting it perform torsional oscilla- . tions. The time T of a torsional oscillation will be =V^' _ B Fig. 30. where I is the moment of inertia of the suspended body about the axis of twist, and N is the moment of the torsional couple per radian of twist, that is the value of P when </> = 1, i.e. XV MODULUS OF RIGIDITY 87 Hence r=27r and n = SttI 11. j"i n If the suspended body is of some regular form its moment of inertia about the torsional axis can be calcu- lated from its mass and shape. Otherwise we may proceed as in the Exercise " Moments of Inertia II." p. 73, to determine /. In order to verify the truth of the law ex- pressed by the above equation, suspend a brass cylinder (Fig. 31) by means of three different wires of the same material but different dimen- sions, and determine the times of torsional oscil- lation (Section III.). In the example given below two of the wires are cut from the same coil, the length of one piece being about double that of the other, and the third is equal in length to the longer of the two former wires but is thicker. The diameters of the wires should be determined by the screw gauge at four different points and the means taken. Tabulate the results as follows : — Fig. 31. 2 March, 1899. Steel wires marked C. Length / cms. Mean raditur OOUI. r* I T r« ^- 83-5 41-75 83-5 0165 0165 0295 7-41xl0-» 7-41 „ 75-72 ,. 8-87 X 10-" Ac. Ac. 6-45 4-54 215 41-60 20-61 4-62 3-69 X 10-» Ac. <fec. Mean 88 MECHANICS AND GENERAL PHYSICS XV Take the mean of the values obtained in the last column and substitute in the equation Stt/ I to determine the modulus of rigidity n. The value of / can be found from the weight and dimensions of the suspended body, since the moment of inertia of a cylinder about its axis is the product of its mass into half the square of its radius. Tabulate your results as follows : — 2 March, 1899. Steel wires and cylinder marked C. Mass of cylinder ... ... ... 663 grams Radius of cylinder ... ... ... 2"27 cms. Moment of Inertia (calculated) = 1578 .*. n for steel wire used = 8*5 x 10" dynes per sq. cm. SECTION XVI. VISCOSITY. Apparatus required : Two long capillary tribes, an inverted hell-jar y a small flask, microscope and stage micrometer. If two layei-s of a liquid at a small distance x- apart are moving with velocities Vj and v^, the faster moving layer tends to increase the velocity of the slower, the slower to decrease that of the faster. If S is the surface of contact of the two layers, and F the magnitude of the force exerted by one layer on the other, then X where rj is the " coefficient of viscosity " of the liquid. The object of the present exercise is to determine the value of t) for a given liquid. If a liquid flows along a capillary tube the liquid in contact with the wall of the tube is either at rest or only moves very slowly. We shall assume that it is at rest. The layer of liquid next to the one in contact with the walls moves with small velocity, the layer next to it with a greater, and so on, the liquid at the axis of the tube moving with the greatest velocity. For the same tube, the velocity of the layers will increase more rapidly from the walls to the axis for a liquid of small viscosity than for a liquid of great viscosity. The volume of the liquid which under given conditions flows through the tube will thei-e- fore depend on the viscosity of the liquid. 90 MECHANICS AND GENERAL PHYSICS XVI It may be shewn that the volume V is given by the equation where R = radius of the tube. p = excess of static pressure at the outlet of the tube over atmospheric pressure. rj = coefficient of viscosity of liquid. t = time in seconds. An inverted bell-jar is provided (Fig. 32), having a rubber stopper through which passes a bent tube, to the end of which Fig. 32. an inclined capillary tube of diameter 2R and length I is attached by a piece of rubber tubing. A strip of paper is gummed to the jar about half-way up, and a horizontal mark on it serves to indicate the level of the liquid in the jar. Place the lower end of the capillary tube about 10 cms. XVI VISCOSITY 91 above the table, and under it put a beaker to catch any liquid issuing. Place a thermometer in the bell-jar. Gum a piece of paper to the neck of a 4 oz. flask, and determine the volume up to the paper by weighing the flask when empty and when full of water. Fill the bell-jar with water at the tem- perature of the room, to a level a little above the upper mark on the gum paper. The water will flow through the tube into the beaker. When the level of the water in the bell-jar has fallen to a convenient mark on the paper strip on the side of the jar, remove the beaker quickly and replace it by the flask, noting the time to a second. When the flask is filled to the level of the gum paper replace it by the beaker, notice again the time and mark on the paper strip the level of the water in the bell -jar. Determine the mean static pressure at the outlet of the tube by measuring the heights of the two marks on the bell-jar above the centre of the bore of the tube at the outlet, and taking the mean, or by measuring the heights of the two marks from the table and subtracting the height of the centre of the bore of the tube above the table. If // is the height, p the density of the liquid and g the gravitational acceleration, p = hgp. The diameter of the tube should be found by placing the tube vertical under the stage of a microscope having a scale in the eyepiece, in such a way that one end is in focus. The greatest and least diameters of the tube at one end are thus found in divisions of the eyepiece scale. Reverse the tube and find the diameter of the other end. If the four measurements are nearly equal take the mean, if they differ materially use another tube. To find the value of an eyepiece scale division in cms., remove the tube, place on the stage of the microscope a scale divided into tenths of millimeters, and find how many of the eyepiece divisions are equivalent to ten of the stage scale. Take a wider tube and determine by means of it the viscosity of the salt solution provided. Find the density of the solution by weighing the flask when filled with it up to the level of the paper. 92 MECHANICS AND GENERAL PHYSICS XVI Tabulate your results as follows : — 21 March, 1899. Microscope A. Divisions of eyepiece scale corresponding to 1 mm. = 61 '5. Value of a division = 00163 cm. Weight of flask filled with water =100 grams. Do. empty = 24 grams. Volume of flask =76 c.c. Weight of flask filled with solution =116 grams. .'. Density of solution = 1*21 Viscosity of water. Tube A, Diameters (1)68-6, 68-4) ^o j- r (2) 67-7, 67 31= ^^^^^-"^"^^P^"^" = ]11 cm. Length = 65*5 cms. Vol. = 76 c.c. Pressure = 19-6 cms. water = 1 9230 dynes per sq. cm, Time = 450 sec. Temperature = 11^-5 C. 7;at ir-oC. = 0103. And similarly for the observations with the solution. SECTION XVII. SURFACE TENSION. Apparatus required and stage micrometer. Glass tube and scale, microscope Heat in the blow-pipe tiame a piece of glass tubing, which has been thorouqhly cleaned irmde with water and dried, and when it is soft draw the two ends apart rapidly so as to form a fine capillary tube. Break off the capillary part of the tube, and repeat the drawing-out process on the rest of the tube till half-a-dozen capillary tubes have been obtained, two about 1, two about '7, and two about *3 mm. diameter. Break off from each tube about 20 cms. of the most uniform por- tion. Mount vertically in a small clean glass vessel containing water a clean transparent gra- duated scale. Wet the scale a little and place a tube against it (Fig. 38) ; the tube will stick and will not need supporting. The water will ascend in the tube. When the water has reached its greatest height, raise the tube a few mms., so that the inner wall above the meniscus is wet, and read on the scale the levels of the meniscus and of the surface of the liquid in the vessel. Fig. 88. 94 MECHANICS AND GENERAL PHYSICS XVII In taking the latter reading the level of the surface at a little distance from the scale should be read, since close to the scale the surface is curved upwards, thus: — Read A, not B. Gum a small piece of paper to the tube, 2 mms. above the top of the column. ^ Raise the tube 5 mms., and again take readings. If the diflference of level of the meniscus inside the tube and of the liquid „. . outside is the same as it was previously, the tube is uniform. If there is much difference another tube should be substituted for it. Repeat the observations with two other tubes of different diameters, attaching a strip of paper to each. Now take another vessel, clean it thoroughly, fill it with 20°/o ethyl alcohol, and take observations with the other three tubes, attaching strips of paper as before. Pour the alcohol back into the stock bottle. Tlie radii of the tubes at points 4"5 mms. below the strips of paper must now be found. To do this a microscope provided with a micrometer eye- piece is required. To determine in cms. the value of a division of the scale of this eyepiece, place on the stage of the micro- scope a scale graduated in tenths of a mm., notice how many divisions on this scale correspond to some convenient number on the eyepiece scale, and hence find the value of a division of the eyepiece scale in centimetres. Remove the stage micro- meter, break off one of the tubes 45 mms. below the gum paper by marking it with a file or rough-edged knife, and mount each part with the broken section upwards on a micro- scope slide, attaching them by a little soft wax. Now place one of these mounted tubes on the stage, focus the broken end of the tube, and read its diameter in scale divisions of the eyepiece micrometer. If it is not accurately cir- cular, measure its greatest and least diameters and take the mean. Repeat with the other piece. Take the mean and convert to cms. XVII SURFACE TENSION 95 Do this with each capillary tube used, and calculate the Talue of the surface tension T in dynes per cm. from the equation ^ "" 2 ' where g = value of gravitational acceleration. p = density of liquid. h = height of meniscus in tube above liquid outside. r = radius of tube. Tabulate your results as follows : — 12 February, 1899. Microscope A. 10 divisions of eyepiece scale = 1*67 divisions on stage scale = 167 mm. .'. 1 division of eyepiece scale = 00167 cm. Liquid Tube Reading in cms. at Height cm. Badius h.r. T Meniscus Base eyepiece divisions cms. Water ■1 3 3-66 3-65 6-65 6-64 1-52 1-51 1-53 1-52 2141 2-14/ 512\ 512| 42-5 •071 •152 742 Alcohol 20 V, 4 5 6 Verify the value found for water by the following method : — Clean carefully the thin sheet of glass provided, and sus- pend it by means of the frame provided from the left-hand beam of a simple balance. Adjust it till its under edge is hori- zontal, and place weights in the right-hand pan till the pointer 96 MECHANICS AND GENERAL PHYSICS XVII of the balance reads zero. Fill a clean glass vessel, wider than the strip of glass, with clean water, place the vessel under the strip and raise it gradually till the lower edge of the glass just touches the surface of the water. Push the glass a millimetre down into the water so as to wet the edge, and then place weights in the right-hand pan of the balance till the edge is just torn away from the water. If m grams, in addition to the weight to balance the glass in air, have to be used, and if I is the length of the edge of glass touching the surface, 7ng=:T.2l, or T^^""^^^. Record as follows : — Weight to balance glass in air = '52 gram „ „ „ touching water = 1'59 granivS Difference due to surface tension = 1*07 „ Length of edge touching water = 7'5 cms. Surface tension of water =70 dynes per cm. BOOK III. HEAT. SECTION XVIII. COEFFICIENT OF EXPANSION OF A SOLID. Apparatus required : Tubes of metal and glass, flow of hot water or steam, thermometer and reading microscope. When the temperature of a solid is raised, the solid gene- rally expands, and if the distance between two given points in the solid is l^ at 0° C. and U at f C. we have, to a sufficient degree of approximation, lt = Ui\-Vat), where a is a constant called the coefficient of linear expansion or dilatation. In order to determine this coefficient, a bar of convenient length may be measured at two temperatures ^ and t,. If ^ and /a are the observed lengths, we have as a consequence of the above relation As a is small in the case of solids, we may as a rule with sufficient accuracy substitute the length at the ordinary tem- perature of the room for i,. Thus the coefficient of expansion of metals generally lies between 00001 and "00004, and their linear dimensions will increase therefore by less than a thousandth part of the original length between 0* and 20** If therefore it is not desired to measure a to an accuracy greater than one part in a thousand, l^ may be taken as identical with the length measured at the ordinary tempera- 8. p. 7 98 HEAT XVIII ture of the room, and with rods about half a metre long an ordinary scale graduated to millimetres will be sufficient for the purpose of measurement. The quantity Za — h being small some method of measure- ment suitable for small lengths must be used, e.g. a microscope provided with a graduated scale in the eyepiece. The heating of the material to be tested is carried out very conveniently if it is given in the form of a hollow rod, since hot water or steam may be sent through it, to raise it to the required temperature. Two tubes of about 60 cms. length are provided, one of glass and one of brass ; and are mounted so that steam or hot water may be passed through them. If two reading microscopes are available, two marks may be made on the tubes one near each end, and the displacements of these marks measured after a temperature change which should not be less than 70° C. The apparatus provided gives sufficiently accurate results with one reading microscope only. A notch' is cut near one end of the tube, and this notch with the tube placed in a horizontal position is made to rest on the knife edge of the stand provided. For additional security an arrangement is made by means of which the rod may be clamped above the knife edge. The other end of the rod may rest on the microscope stage, but in the stand provided it is supported by passing loosely through a hole cut into a piece of wood fixed to the stand. Fig. 35. Clamp the stand firmly to the bench or slab on which the experiment is to be performed. The free end of the tube XVIII COEFFICIENT OF EXPANSION OF A SOLID 99 carries a narrow paper label with a fine pencil line, drawn at right angles to the tube. You are provided with a microscope having an objective of low power so that it is capable of including a field of view of about 2 mm. diameter. The eyepiece of the microscope carries a micrometer scale. Place the microscopes in such a position that the pencil line is near the centre of the field and then clamp the microscope firmly to the table or slab (Fig. 35). Attach rubber tubes to the ends of the experimental tube and send a stream of cold water through ; observing the tem- perature of the water by means of a thermometer placed in the issuing stream. Observe the reading in the microscope, turn- ing the scale until its divisions are parallel to the pencil line. Send steam or hot water through the tube and again take readings. Finally repeat the experiment with cold water. Substitute the second tube and test similarly. Measure the length between the notch and the pencil line to the nearest mm. Determine the value of a scale division of the eyepiece by placing a scale graduated in -J^ mm. on the stage of the microscope section. Record and calculate the coefficient as follows : Date, June 8, 1899. Microscope A. 58 divisions of eyepiece scale = 10 on stage scale = one millimetre .'.1 division of the eyepiece = 00172 cms. Brass tube 58*6 cms. between marks. Temp. Reading Difference Scale Cms. Water 30' C. Steam 100 Water 30 1-42 5-72 1-42 4-30 •0744 •0744 • • ^ 68-6 X 70 Similarly for the glass tube. •0000181. 7—2 SECTION XIX. THERMAL EXPANSION OF A LIQUID. Apparatus required : Bulb with graduated stem, and ice. The change of volume of a liquid with increase of tempera- ture is most conveniently found by enclosing the liquid in a vessel made of a material of which the coefficient of expansion is known, and observing the apparent change of volume of the liquid as the temperature of liquid and containing vessel is varied. Since the apparent expansion is, in general, only a small fraction of the total volume, the volume of that part of the vessel in which the expansion is measured ^ should only be a small fraction of the total volume. o The vessel, which is called a " dilatometer," is generally a glass bulb provided with a graduated stem having a fine bore, the liquid filling the bulb and part of the stem (Fig. 36). If at 0° C. the surface of the liquid in the stem stands at the division Uo of the stem, and the volume of the bulb up to the zero division of the scale is m, measured in scale divisions, the volume of the liquid at 0'' C. is m + tiq scale divisions. If when the dilatometer and liquid are at f C. the surface ,of the liquid reads n on the scale, we have, for the contents of the vessel up to the mark n at tem- „. ^ Fig. 36. perature f C, {m + n){l + at), where a is the coefficient of expansion of the vessel. If further fft is the expansion of unit XIX THERMAL EXPANSION OF A LIQUID 101 volume of the liquid between 0° and i°, the volume of the liquid at the latter temperature will be (m + n©) (1 + I3t), hence : (m + n) (1 + at) = (m + Uo) (1 + A) orl+A = -^^(l+«0 '^ m + 7?o = 1 + ^ -\- at approximately. m from which the expansion /8f may readily be determined. Unit volume at the freezing point of water will occupy a volume 1 + ^^ at temperature t. A graduated capillary tube is provided, at the end of which a bulb of about 1 cm. diameter has been blown. Wash and dry the bulb and tube and then weigh. Heat the bulb gently in a bunsen burner, then dip the open end of the tube into a small quantity of the liquid to be tested which, if necessary, has been boiled to free it from dissolved air. As the air in the bulb contracts, liquid is drawn up the tube into the bulb. Repeat the heating and cooling till the bulb and a short length of the stem are filled with liquid. Place the tube in a water bath at about 16° C, and after waiting 10 minutes to allow the liquid in the bulb to take up the temperature of the bath, read the temperature of the bath and the position of the liquid in the stem. Weigh the bulb and contents. The difference between this weight and the previous one is the weight of the liquid in the bulb, and if its density is known, the volume of the liquid, therefore the volume of the bulb and stem containing it, can be found. To determine the volume of one scale division of the stem, heat the bulb gently till the liquid stands rather more than half way up the stem, then plunge the open end of the stem into clean mercury, which will be drawn up the tube as the liquids contract. Allow the dilatometer to cool till the posi- tions of both ends of the mercury thread can be read on the scale. 102 HEAT XIX Observe the length of the thread and weigh the bulb and its contents again. The difference between this weight and the last one is the weight of mercury, which may be divided by 13'.596 to give the volume, and hence the volume of a scale division in c.c. may be found. The dilatometer should again be heated till the mercury thread is expelled from the tube, and the upper end of the tube then sealed in the blowpipe (Fig. 36). A label is attached to the tube giving the mass of liquid in the tube and the value of a scale division, from these numbers the quantity m may be calculated. If the liquid is water particular care must be taken with the observations between 0° and 10° owing to the anomalous behaviour of that liquid. The dilatometer should first be placed in a bath containing melting ice, and after the position of the liquid has become constant, a reading should be taken. The temperature of the bath should then be raised about two degrees, and observation of temperature and apparent volume taken at the new temperature. The observation should be repeated at intervals of 2° C. up to 10° C. and then at intervals of 5° C. over the range of temperature desired, and again as the temperature is decreased. The observation and results should be tabulated as follows : Expansion of Water. 10 February, 1899. Dilatometer A, m = 15500, a = '000020. Temp. Reading n-iiQ n — iiQ m at ^t 0°C. 24-0 0-6 23-2 - -8 - 0000516 •000012 - -000040 1-7 22-0 -2-0 -•000129 -000034 - -000095 3-3 20-4 -3-6 - -000232 •000066 - -000166 5-3 2M -2-9 - -000187 •000106 - -000081 7-3 22-2 -1-8 -•000116 •000146 + -000030 12-3 28-0 + 4-0 + -000258 •000246 + ^000504 &c. &c. &c. XIX THERMAL EXPANSION OF A LIQUID 103 Numerical calculation will be facilitated, if the reciprocal of m is first calculated and then multiplied successively by the numbers in the thiid column. Three figures are sufficient and Crelle's Tables may be used. A curve (Fig. 37) should be drawn with the observed temperatures as abscissae and total volumes 1 + y9< as ordiuates. — r " 1003 .' l-OOl y '' - „* *' -• 1000 t '' '- '♦<. ^ _,, i' ' 1 0' 2 0' _ Fig. 37. The numbers, giving the change of volume of water, should be taken out of some book of physical constants, and repre- sented by a second curve for comparison. Care must be taken to wait sufficiently long after each change of temperature to allow the water in the bulb to come to its final state, for it must be remembered that the thermal conductivity of water is small, and that convection currents are diminished in efficacy owing to the small size of the bulb. The numbers taken with rising and falling thermometer should be plotted separately and should shew no systematic difference. SECTION XX. COEFFICIENT OF INCREASE OF PRESSURE OF A GAS WITH TEMPERATURE. Apparatus required : Bulb and pressure apparatus, water bath, thermometer. If the temperature of a constant volume of gas is raised, the pressure of the gas varies according to the equation ^j = p^ (1 + ^t), where po is the pressure at 0° C, t the tem- perature Centigrade, and /S a constant, called the coefficient of increase of pressure with temperature. Fig. 38. XX INCREASE OF PRESSURE WITH TEMPERATURE IN AIR 105 To verify the law expressed by the above equation in the case of air the apparatus shewn in Fig. 38 is provided. A is SL glass bulb to which a tube of small bore BC is attached, i) is a similar tube. The lower ends of the tubes are joined together by a piece of rubber tubing which can be compressed by turning a screw which acts on the plate E. The rubber tube and the lower part of the tubes C, D, are filled with mercury. By adjusting the quantity of mercury and the pressure at E the surface of the mercury in the left-hand tube may be brought to the fixed mark C, If the position of the surface of the mercury in D is then read, and the height of the barometer is known, we have the total pressure to which the air in A is subjected. To determine how this pressure varies with the temperature of the air in the bulb, fill the bath in which the bulb is placed with cold water, noting its temperature. Adjust the mercury to C, and read the positions of D on the scale along- side. Raise the temperature of the bath about 5° C. keeping the water stirred, and repeat the adjustment and readings. Take four observations at intervals of 5° C. and then wait until the thermometer has risen 25°, i.e. stands 40° higher than at the beginning of the observations. Take four more observations at intervals of 5°. Then release the pressure of the screw so as to lower the mercury in the tube C, and pour sufficient cold water into the bath to lower its temperature 5° C. Stir well and raise the mercury to C again, then read D. Lower the mercury, cool further and repeat the observations till the original tem- perature is reached. Take the means of each pair of observations at nearly the same temperature and use them in what follows. Read the barometer ; the pressure of the gas in terms of a column of mercury will be the sum of the height of the barometer and the diflference in level between C and D. If p is this pressure and v the volume of the gas at temperature t, we have pv = p^Voil ■¥ ^t), where Vo is the volume at 0°C. ; t; will increase slightly as the temperature rises owing to the expansion of the glass. 106 HEAT XX If the coefficient of expansion of the glass is a, the volume of the vessel v is obtained from v = Vo(l + at). Hence P = ^«H2- Or since at is, in the above experiments, always less than 002, we may write p=Po(l-\-{^--oi)t). If piti, pit^ are two sets of readings of p and t, we may form two similar equations and eliminating ^o find Pi ^ Pi (l+(/3-a)^,) H-(/3-a)^' and hence p^ Pi /8 may therefore be calculated, if a is known, by observation of any two corresponding pressures and temperatures. If a number of readings are taken at regular intervals of tempera- ture we may apply the method explained in Section (III) and secure the smallest probable error by taking a certain number of observations, omitting the same number, and finally taking again an equal number. The observations are combined and reduced as given in the Table below. Of the two forms for (/S — a) given above the first involves less arithmetical labour, and would naturally be taken in the case that the pressure is observed only for two temperatures. But for a series of experiments the results may be tabulated in a more sym- metrical manner if the second form is adopted. The labour of calculation is reduced if the reciprocals of pi and jh are obtained from Barlow's Tables. The fraction — etc. as well as the final quotient may be Pi found with sufficient accuracy from Crelle's multiplication tables. XX INCREASE OF PRESSURE WITH TEMPERATURE IN AIR 107 Arrange the observations as follows : Apparatus marked K. 8 Nov., 1899. Height of Barometer 76*2 cms. 1 'i 1 h 1 1 t- t^ h. Px h I'a — /3-0 Pi l>i P% Pi Pi Pt Pi Pi 10°-5 77-7 •01287 •135 60-3 88-4 •01131 •567 •00156 •432 •00361 15-2 78-9 1267 •191 55-1 89-6 1116 •609 151 •418 361 20 801 1248 •249 60 90-9 1100 •660 148 •411 1 360 26-0 81-3 1230 •307 65-0 92-1 1086 •706 1 144 •399 361 Mean value of (y9 - a) For glass of bulb a . . . yS for air 1 fi = -003608 = 00002 8 = 003636 = 275° According to Regnault fi should be equal to '0003669. There are two sources of error both tending to reduce the apparent value of fi. One is the expansion of the glass vessel due to increased internal pressure at the higher temperatures, and the other is due to the fact that the air in the tube leading from the bulb is not at the same temperature as that in the bulb. The first named error is not likely to be more than •000003 with the dimension of apparatus used, the second error depends on the width of the bore, which should therefore be narrow. If the bulb is not properly dried, too high value will be obtained owing to the evaporation of the water from the walls of the vessel. The increase of pressure of air is used to measure tempera- tures in the so-called "air thermometers," the construction of which is the same in principle as that of the apparatus used in this exercise. SECTION XXI. COEFFICIENT OF EXPANSION OF AIR AT CONSTANT PRESSURE. Apparatus required : Graduated tube and index, water bath, and thermometer. When the temperature of a gas maintained at constant pressure is raised, the volume increases, and may be repre- sented by an expression of the form v = Vo(l+at), where v is the volume at ^°C., Vo at 0°C. and a is a constant called the coefficient of expansion of the gas. If the gas is enclosed in a vessel the coefficient of cubical expansion of the material of which is /3, the apparent volume of the gas will be expressed by the equation v = Vo(l+a^l3t). If ti and t^ are two temperatures and Vi, v^, the corresponding apparent volumes, it is found eliminating Vq that 1_1 _ _ Hence if observations of the variation of volume of a given mass of gas, enclosed in a vessel of known coefficient of expan- sion, are made, the coefficient of expansion of the gas may be determined. The apparatus provided consists of a graduated glass tube of fine bore, one end of which is sealed, and the other attached XXI EXPANSION OF AIR AT CONSTANT PRESSURE 109 to a wide tube bent at right angles. The complete tube is attached to a brass frame. The capillary tube is filled for about two-thirds of its length with air, which is confined by a thread of strong sulphuric acid. This acid thread also serves to indicate the volume of the air and to keep it dry (Fig. 39). The wider part of the tube serves for introducing the acid, so as to enclose a convenient quantity of air. Fig. 39. Fig. 40. To find the eflfect of change of temperature on the enclosed air, place the graduated tube horizontal in a water bath the temperature of which can be varied (Fig. 40). Mix ice and if necessary salt with the water till the temperature indicated on a thermometer placed in the water is 0°C. Stir the water well and aft^r waiting till the thread of acid has taken a constant position, take a reading of that end of it which is in contact with the gas. Increase the temperature of the water by 5° C, and again observe the volume. Take four observations increas- ing the temperature in steps of 5^ C, then wait until the tem- perature has risen another 25° and take four more observations at temperatures increasing by 5°C. Then decrease the temperature and take observations at approximately the same temperatures as during the rise. The closed end will not in general be at the zero of the 110 HEAT XXI scale and its reading must therefore be subtracted from the readings of the index to obtain the volumes. As the closed end is often conical a short thread of acid may be introduced to occupy the conical spaces. It is assumed that the bore is uniform so that the differences in the readings of the scale are proportional to the volumes. The observations are reduced as in the previous section. Enter as follows : Apparatus marked B. 12 November, 1899. Reading at closed end 5*05. Temperature increasing Temperature decreasing t Beading Vol. t Eeading Vol. 0-0 5 10-1 15-1 15-25 15-42 15-59 15-80 10-20 10-37 10-54 10-75^ •0 4-8 10-2 15-1 15-25 15-42 15-61 15-76 10-20 10-37 10-56 10-70 39-5 45-0 49-8 54-6 16-67 16-83 17-03 17-23 11-62 11-78 11-98 12-18 40-5 45-2 50-6 55-6 16-71 16-87 17-07 17-27 11-66 11-82 12-02 12-22 Take the mean of the numbers for increasing and decreasing temperatures and enter as in the following table. 1 h 1 h 1 1 ig «l tl ^1 — «2 ^2 — a-/3 ^'i ^1 ^2 ^2 ^^1 ^2 V2 Vi o°c. 10-20 -0980 40° -0 11-64 -0859 3-436 -0121 3-436 -00353 4-9 10-37 •0964 •472 45-1 11-80 -0847 3-819 -0117 3-347 •00349 10-15 10-55 -0948 -962 50-2 12 00 -0833 4-182 -0115 3-220 •00357 15-1 10-72 -0933 1-409 55-1 12-20 •0820 4-520 -0113 3-111 •00363 Mean value of a — yS ^ for glass used .*. a = -00355 = -00003 = -00358 XXI EXPANSION OF AIR AT CONSTANT PRESSURE 111 For gases satisfying Boyle's law the coefficient of expansion at constant pressure is the same as that of increase of pressure for constant volume. The value of a found is about two per cent, too small. Take temperatures as abscissae, and volumes in scale divi- sions as ordinates and draw a straight line passing as nearly as possible through the points so determined (Fig. 41). From the inclination of the straight line find the value of a — yS. |0,ri_ , ^ ^ \^ f^ ^ v..) ^ -^ ^ ^ '^ 1 1 rk .. '^ 1 ^ ^ ^ <^ w^ ^ ■^ 10 20 T>4Dp 30 Fig. 41. 40 60 SECTION XXII. EFFECT OF PRESSURE ON THE BOILING POINT OF A LIQUID. Apparatus required : Thermometer, flask, condenser and air-pumps. When the pressure under which a liquid boils is increased, the temperature to which the liquid must be brought before boiling takes place is raised, and if the pressure is decreased the temperature at which boiling occurs is reduced. It is the object of the present exercise to find the relation between the two changes. In the diagi-am of the apparatus to be used (Fig. 42), ul is a Bunsen filter pump through which a rapid flow of water from the tap T^ may be kept up. Fig. 42. XXII RELATION BETWEEN PRESSURE AND BOILING POINT 113 (7 is a mercury manometer, with a reservoir R for adjusting the reading of the surface in the open tube to zero of the scale alongside. i5 is a trap arranged so as to prevent the water from the pump flowing back to the manometer when the pump is stopped. i) is a three-way cock having a central passage parallel to the handle, this passage joins at the centre to another passage at right angles leading only to one side. The side to which it leads is marked by a black dot. When the handle is vertical with the black dot to the right, the pump is in connection Mrith the condensing apparatus. Call this position A. When the black dot is to the left the apparatus is open to the air. Call this position B. When the handle is horizontal with the black dot below, the way to the bottle is closed, and the pump is in connection with the air. (Position C.) J5^ is a stop-cock to cut off the boiling apparatus from the pump. ^ is a bottle to diminish the changes of pressure due to the bursting of bubbles in the flask N during boiling. r is a thermometer giving the temperature of the steam rising from the liquid experimented on — water in this case. J/ is a condenser in which the steam is condensed, the condensed liquid running back into the flask. To commence the experiment, put the flask N in connection with the air by turning D till the handle is in the position B. Set the water in the flask boiling, and see that water is passing slowly through the condenser from the tap T^. Read the barometer, and calculate the true boiling point of water at the observed pressure, taking the boiling point as changing '37° C. per cm. change of pressure. When the temperature is steady, read the thermometer T, and determine the error at the boiling point. The whole of the mercury thread should be inside the flask. If a considerable portion of it lies outside a correction called that of " emergent stem " becomes necessary. If the error of the thermometer at zero has been previously determined and marked on the thermometer, the correction at any reading may be found with sufficient accuracy for the present purpose by drawing the curve of corrections as in the Intermediate Course of Practical Physics, 8. p. 8 114 HEAT XXII Now turn D into position A, and set the pump working. When the difference of level of tbe mercury columns in G is 4 cms., turn the stop-cock into position C, turn off the water at tap Tj, and, after waiting two minutes, read the thermometer and manometer three times at half-minute intervals. After taking the observations, turn on the water again, place D vertical with the dot to the right, and diminish the pressure 4 cms. further. Take readings as before. Repeat the observations at intervals of 4 cms. pressure till the temperature is about 75"; then with the pressure increasing again to that of the atmosphere. Take the mean of each set of three, correct it for the errors of the thermometer, and tabulate your results as follows : — 1 April, 1899. Observed height of barometer ... Calculated boiling point Observed reading of thermometer atmospheric pressure Hence correction of thermometer at boil- ing point ... 763-2 mms. ... 100°-1 C. at vil 99-3° )ll- = -h-8° Pressure Temperature at manometer in flask Observed Correction Corrected 40-2 78-1 121-2 162-1 &c. 763-2 723-0 685-1 642-0 601-1 99-3 97-8 96-3 94-6 92-8 + •8 •8 -8 •8 •8 100-1 98-6 97-1 95-4 93-6 <fec. Now plot a curve with pressures as abscissae, and the corrected temperatures as ordinates, representing the points obtained with the pressure decreasing by a cross, those with the pressure increasing by a circle. The two sets of observations should not differ by more than about '1° from each other. If they do, sufficient time has not XXII RELATION BETWEEN PRESSURE AND BOILING POINT 115 beeD allowed between each observation for the temperature of the steam to become constant. From the observations calculate the temperatures corres- ponding to pressures of 760, 720, 680 mms., etc., by graphical construction or as follows : — Let pi be the observed pressure which is nearest to the pressure p for which the temperature t of boiling is to be determined. Let p^ be a second observed pressure near p. Then if ^ and ^2 be the temperatures of boiling corresponding topi.p^ Pi p\ The second term will be less than one degree, if the observations have been taken according to the above in- structions. Hence this term need only be calculated to one significant figure. Tabulate the results along with Regnault's as follows : — Pressure Temperature Experiment Begnault 760 mms. 100-0 100° OC. 720 98-6 98-5 680 96-9 640 95-2 600 93-5 560 91-7 520 89-7 480 87-6 440 85-4 400 830 360 80-4 320 77-5 8—2 SECTION XXIII. HYGROMETRY. Apparatus required bulb thermometers, ether. DanielVs hygrometer, wet and dry The most important determination in connection with the aqueous vapour present in the atmosphere is that of the "relative humidity" or "fractional saturation," i.e. the ratio of the amount of moisture actually present at any time in a given space, to the amount which would at that temperature saturate the space. This quantity may be determined directly by passing a known volume of moist air through tubes filled with drying materials and finding the increase in weight of these materials due to the moisture absorbed. It is determined in practice however by one of two methods : — dew point observa- tions, or wet and dry bulb observations. In the first of these methods the air is cooled down till the vapour present in it begins to condense, and the temperature at which condensation begins is observed. The instrument used is called a dew point hygrometer. The form known as Daniell's hygrometer consists of a tube with a bulb at each end containing ether and vapour of ether only. The outer surface of the lower bulb is silvered, gilded or made of black glass and encloses a thermometer ; the upper bulb is covered with muslin (Fig. 43). A thermometer fixed to the stem indicates the temperature of the air. Determine the dew point at a quiet spot outside the laboratory, in the following way. Tilt the instrument till the Fig. 43. XXIII HYGROMETRY 117 ether has run into the lower bulb. Pour a little ether on the muslin surrounding the upper bulb. As this ether evaporates the bulb is cooled and the ether in its interior condensed. The ether in the lower bulb now evaporates and is cooled. Watch carefully the polished surface of the lower bulb to see when moisture is first deposited on it. Keep the muslin saturated with ether, and occasionally shake the instrument so as to stir the liquid in the lower bulb. Note the temperature of the air and of the thermometer in the bulb, when moisture first appears. The instant of appearance may be best detected by watching the image of some object in the polished surface of the bulb. Cease appl3dng ether to the muslin and watch when the last sign of moisture disappears. Note again the temperatures and take the mean of the readings in each case. The mean temperature indicated by the thermometer in the bulb is the dew point. Find from tables the pressure of aqueous vapour at the dew point, and at the temperature of the air. The quotient is the relative humidity, or fractional saturation. Record as follows: — 16 March, 1897. Temp, of air = 17*0° C, .*. pressure of vapour = 1*44 cms. of mercury „ dew point = 12*5° „ „ =1*08 cms. of mercury Relative humidity = — j^; = '*^^- The second method consists in observing the difference of readings of two thermometers, one with its bulb surrounded by a wet wick and the other with its bulb free. The arrangement is known as the wet and dry bulb thermometer (Fig. 44). The indications of the wet bulb do not depend, however, exclusively on the hygrometric state of the air, for if the air near the t hermometers is in motion, evaporation is more rapid and the reading lowered in consequence. The readings can 118 HEAT XXIII only be interpreted by means of tables based on experiments, but it is found by comparison with more accurate hygrometers, that if the air round the thermo- meters is neither absolutely still nor violently disturbed, these tables give results which are sufficiently accurate for meteorological purposes. Saturate the muslin surrounding the wet bulb of the hygrometer provided, and the wick hanging from it, with water, and place it in a position so that the air can flow across the bulbs of the thermometers. If the air is still, suspend the instrument by a string from a fixed point, and set it slowing swinging as a pendulum. The temperature of the wet bulb will be found to decrease. When it becomes steady, read both the thermometers, and use table 35, Lupton, p. 30, to find the pressure of the aqueous vapour present in the air. Then divide by the pressure of saturated vapour at the temperature of the dry bulb. The quotient is the relative humidity, or fractional saturation. Record as follows : Temp, of air = 17° C, .*. pressure of vapour = 1*44 cms. „ wet bulb = 14-5° „ „ „ =1-09 „ 109 Relative humidity = r-^^T = *'^^- Fig. 44. SECTION XXIV. LAWS OF COOLING. Apparatus required: Calommeter, with (a) simple en- closure, (b) enclosure luith water jacket ; glass tubing to be used as pipette; thermoineter graduated in -^ degrees and reading frorti about 10° to 25° C; thermometer graduated in -^ degrees and reading up to 50° C. If a body differs in temperature from its surroundings, it will lose or gain heat by radiation and by conduction to the bodies surrounding it. The conduction is much affected by air currents caused by the air in contact with the calorimeter ascending and being replaced by colder air. Conduction is thus increased by the fresh supply of cold air, and is rendered irregular if by any cause the movement of the air is accelerated or retarded. Heat earned away by the bodily motion of the air is said to be carried by "convection." Experiment I. To determine the rate of cooling of a calorimeter under various conditions. Draw a pencil line round the inside of the calorimeter (Fig. 45) provided, about 1 cm. below the rim. Fill it to the mark with water at a temperature about 20° above that of the room. Place the calorimeter on the table unprotected, and place a thermometer, reading to 50°, in the water. Take readings of the temperature of the water every half minute during a period of two minutes, stirring the water well the whole time, and, estimating to 0^01 C, wait two minutes and then take live further observations at half minute intervals. Take the temperature of the air by means of the smaller tliermoraeter in the interval of waiting. 120 HEAT XXIV Kaise the water to its original temperature, and repeat the experiments with the calorimeter protected by an outside vessel but not covered ; and thirdly, with the calorimeter covered. CI=3> Fig. 45. The experiments are entered and reduced as follows : 12 Oct. 1898. 1. Calorimeter unprotected. Time hr. min. Temperature Time hr. min. Temperature Fall in 4 min. 3 10-0 10-5 110 11-5 12^0 38°^18C. •10 •00 37-94 •86 3 140 14^5 150 15^5 16-0 37°-53C. •46 •37 •29 •23 •65 •64 •63 •65 •63 Means 38^016 37-376 •640 Mean temp. 37° •696 Average fall in four minutes „ per minute Temperature of air •640 •160 18°-5 C. XXIV LAWS OF COOLING 121 'ti The rates of cooling when the calorimeter is protected by an outside vessel, and covered or uncovered, must be reduced in exactly the same way. Compare the results in the three cases, and draw your conclusions. Experiment II. Having seen how the effects of con- vection currents can be diminished by properly protecting the calorimeter, we must next study how the amount of the cooling, in nearly quiescent air, varies with the temperatures of the body cooled and of its surroundings. You are provided with a calorimeter which can be placed in an enclosure consisting of a vessel having a double wall, into which water at the temperature of the room may be poured (Fig. 46). Two thermometers graduated in tenths give the temperatures of the water in the calorimeter and enclosure respec- tively. Fill the calorimeter with water up to the pencil line drawn round the inside 1 cm. below the rim. Raise the water to a tem- perature approximately 5° above that of the room, and observe the temperature every half minute for two minutes, then wait two minutes and observe again for two minutes, keeping the water well stirred. Note the temperature of the enclosure, which should be approximately the same as that of the room, at the beginning and end of the observations. Tabulate as shown previously. Raise the water in the calorimeter to temperatures 10°, 20°, and 30° above that of the room, and observe the rate of cooling in each case, adjusting the temperature of the enclosure to be the same within a few tenths of a degree during each set of observations. Calculate in each case the ratio of the rate of cooling to the excess of the mean temperature of the calorimeter over that of the enclosure. Call that ratio k. I. XZZ3Z Fig. 46. 122 HEAT XXIV Collect your results as follows : Rate of Cooling per min. Mean temperature Mean temperature Cooling per per degree excess of calorimeter of enclosure Difference minute =h 46-54° C. 14-30° C. 32-24° C. -494 -0153 37-31 14-25 23-06 -281 0122 27-58 14-45 13-13 -158 -0120 19-66 14-40 5-22 -056 -0107 Mean -0126 The quantity called h is the rate of cooling for a difference of temperature of one degree between calorimeter and enclosure, calculated on the assumption that the cooling is proportional to the difference of temperature between the calorimeter and enclosure, an assumption which the result shews is only ap- proximately correct. If the mass of water in the calorimeter is M, then the heat lost will be Mh gramme degrees per minute, and if A is the total area of the calorimeter exposed, the heat lost per unit surface in one minute will be MhjA, or if k is the heat lost per sq. cm. of surface per second, per degree excess of temperature Calculate k from the mean value of h found, and enter your results as follows. For small differences of temperature k should only depend on the nature of the radiating surface. Inner radius of enclosure ... ... ... 6 cms. Weight of water in calorimeter ... ... 146 grms. Radius of calorimeter ... ... ... 3*2 cms. Height of „ ... ... ... 5"1 „ Area of curved surface of calorimeter . . . 102'6 sq. cms. „ top and bottom „ ... 64'4 „ Total area of radiating surface ... 167"0 „ Mean value of h found above = "0126. ^ , 0126 X 146 .^^^i«^ ^^^^^ ^= 60x167 =QQQ1^^' The inner radius of the enclosure is not required in the calculation, but should be measured and recorded, in order that XXIV LAWS OF COOLING 123 the effect of the dimensions of the air space on the heat lost by conduction and convection might be traced by comparing the results obtained under different conditions. Newton's Law of Cooling states that the rate of loss of heat of a hot body is proportional to the difference in tempera- ture between the surface of the hot body and the surrounding space. The above experiments have shewn that the law is only approximately true, but it holds with sufficient accuracy in heat measurements where the differences in temperature do not exceed a few degrees. In order that the results obtained in the above experiments may be utilised in calorimetric measurements, a few further explanations are necessary. If h is the cooling in a given interval of time of a calorimeter containing a liquid, the specific heat of which does not vary with the temperature, when the difference in temperature between the calorimeter and the enclosure is one degree, Newton's law states the rate of cooling H when the tem- perature of the enclosure is T^ and that of the calorimeter r to be H^h{T-T,). If Tq is known, the value of h may be determined from a single experiment, as in Exercise II. above. It is often impossible to know the value of T^ accurately, as different parts of the enclosure may have different temperatures. In this case the above equation will still hold, but Tq will represent the mean temperature of the enclosure. To deter- mine h we then require two sets of observations with different values of T, but the same T^. Let H^ be the cooling of the calorimeter in a certain interval when the temperature is 2^, and H^ when the temperature is T^ ; then H,^h{T,-T,) and H^^h{T,--T,). As If,, ifg, Tx, T^ are known quantities, the values of h and T^ may be deduced and substituted in the general equation H^h{T"T,), 124 HEAT XXIV It is found in this way that and ,,^^^-r)-^.(r.-r) which gives the fall of temperature in the interval, for any value of T. Sufficiently accurate values may be obtained by a graphical method. Let OAo (Fig. 47) represent the temperature To of the enclosure, and OAi the temperature Tj of the calorimeter for which the cooling has been determined. Let AiN^ represent the rate of cooling measured on some suitable scale. Join -^o-A^i- Then if OA be any temperature T, the corresponding rate of cooling will be measured by AN. For AN AAo _ jT — To _ Rate of cooling at temperature T AiNi ~ AjAq ~ T^ — Tq ~~ Rate of cooling at temperature T^ ' Hence, if a curve is drawn such that the ordinates measure rates of cooling, and the abscissae the corresponding temperature of the cooling body, the curve is a straight line within the limits of accuracy of Newton's Law of Cool- ^-"^^^o ing. Fig- ^7. If the temperature T^ is not known, but the rates of cooling H^ and H^ are found for the two temperatures T^ and Tg, we may deduce the rate of cooling for a temperature T as follows : Take OA^ and OA^ (Fig. 48) to represent T^ and T^, A^N^ and A^Nc^ the rates of cooling H^ and H^. Join N-^N^^. If OA be any temperature T, then AN will be the corresponding rate of cooling. It may happen that at the beginning of an experiment, the water in the calorimeter at a temperature T^ does not fall but rises in temperature. In that case the line AiN-^ must be measured downwards as in Fig. 49, i.e., it must be taken as negative cooling. In drawing the curves, attention should be given to the XXIV LAWS OF COOLING 125 scale. If the rates of cooling are measured to one per cent., for instance, which in all cases will be sufficient, the length of Fig. 49. the line A^N^ should be such that the hundredth part of it may be estimated, i.e., the unit in which it is measured should be chosen so that A^N^ is at least equal to ten times the length of a division on the squared paper used. The scale of temperatures may conveniently be such that the angle between A^A^ and N^N^ is between 30° and 45°. From the results obtained in Exercise II., making use of the observed rates of cooling in the two cases in which the differ- ences in temperature between the enclosure and calorimeter is least, determine, by the graphical method just described, the temperature of the enclosure, supposed not to be known. Thus, in the above example, one would take the abscissae proportional to 27*6 and 197 and the ordinates in some convenient scale proportional to 158 and 56. The line drawn through the two points so determined, intersects the axis of temperatures at the point representing the temperature of the enclosure. Carry out the construction for your own experiments, record the value of To thus found, and compare it with that obtained by direct observation. In order that jTo should be capable of being determined independently, it is clearly necessary that it should not vary during the course of the experiment. This is the reason why in Exercise 11. the temperature of the enclosure was adjusted so as to be the same in each case. SECTION XXV. COOLING CORRECTION IN CALORIMETRIC MEASUREMENTS. Apparatus required : Calorimeter with water jacket, ther- mometer graduated in tenths of degrees, india-rubber and heater. This exercise is an application of the principles explained in the exercise on the " Laws of Cooling," to the determination of the specific heat of india-rubber. India-rubber is a bad conductor of heat, and in consequence gives up its heat to the calorimeter at a slow rate; hence the correction for cooling is of special importance. The object of the exercise is not so much to obtain an accurate value of the specific heat as to shew the principle of reducing calorimetric observations. Place the india-rubber in a heater, and raise it to about 100° C. This may be done with sufficient accuracy by placing it in boiling water for not less than a quarter of an hour. While the india-rubber takes up the proper temperature, weigh the calorimeter and fill it and the water jacket with water which has previously been standing in the room, so as to take up its temperature — tap water is generally much colder. Weigh the calorimeter again to determine the quantity of water in it. Place the calorimeter with the thermometer in it in its water jacket, and leave it for two or three minutes before taking any observations. In the meantime prepare your note- book by drawing vertical lines to allow for five vertical columns. The first column is reserved for the insertion of the time at which an observation is taken, which should be carefully put down; the second is for the temperature observed. XXV COOLING CORRECTION 127 The experiment is divided into three stages, during which observations of temperature are taken every half minute. 1. The initial period, before the introduction of the india- rubber into the calorimeter. This should last six minutes, and should only be begun when the change of temperature is small and steady. Observations are taken at half-minute intervals during two minutes, then stopped during two minutes and resumed again. 2. The principal period, in which the india-rubber is in the calorimeter. The thermometer will be found at first to rise, then to remain nearly steady, and then to fall. When the fall has become uniform we arrive at : 3. The final period, of six minutes, during which the fall is steady. Observations are taken as in the first period. About 20 seconds after the last reading of the first period, the india-rubber is removed from the boiling water, the adhering water shaken off, and at the half minute the rubber is lowered into the calorimeter. During the ten seconds that the india-rubber is exposed to the air the water on its surface evaporates, and is not carried into the calorimeter. In an accurate measurement of specific heat this would not be a permi.ssible process, as the parts of the india-rubber exposed to the air are cooled by evaporation of the water and by conduction to the air. The substance would have to be heated in an air chamber kept at 100° C. by being surrounded by steam, and be suddenly dropped into the calorimeter. When the india-rubber has been placed in the calorimeter, the thermometer is observed at the next half minute and each succeeding half minute. The first few temperature observations will only be approximate, as the temperature will be rising quickly. Observation of temperature should be made till it has ceased to rise and has fallen at a constant rate for six minutes. In order to reduce the observations and calculate the specific heat, deduce from the observations taken during the first and third periods the rates of cooling or heating at the mean 128 HEAT XXV emperatures of the two periods, and determine graphically, as explained in the previous section, the rate of cooling corre- sponding to any temperature of the calorimeter. Enter^ in the third column of the note-book the mean temperature of the calorimeter during each interval of time throughout the experiment. Determine from the construction and put down in the fourth column, the diminution of tempe- rature owing to the loss of heat during each interval. In the fifth column tabulate the total loss in temperature up to each time given in the first column. An inspection of the table below will shew how this column is obtained. In the example given the loss was "005 to the end of the first interval, and during the second interval the loss was "009 ; the total loss up to the end of the second interval was therefore "014, and if to this is added the loss during the third interval we obtain '025 and so on. The last column, which gives the temperature which the calorimeter would have shewn if there had been no loss of heat due to radiation and conduction to the air, is obtained by adding the numbers in the fifth column to those in the second. It will be seen that the temperatures given in this column rise at first quickly, then slowly, and during the last period remain sensibly constant. The mean corrected temperature during that period would therefore have been 22*54'', if there had been no loss of heat. First Period. 13 March, 1899. Temperature slowly falling. Time Temp. Time Temp. Cooling ll'^OO-O'" 19°-69 111^4-0™ 19°-67 •02 0-5 ■69 4-5 •67 •02 10 •69 5^0 -m 1 -03 1-5 •68 5-5 •66 1 ^02 2 •67 6-0 •66 •01 Means 19*'-684 19°-664 •020 XXV SPECIFIC HEAT OF INDIA-RUBBER 129 The mean of 19*684 and 19664 gives the Temperature at ll*' 3°» = 19°-674 Cooling in 4 mins. = '020 i .. ='0025 3i „ = 0°018 Temperature at ll'^e^'S) _ jqo.^-p when rubber introduced] ~ Second Period.* Temp. Average temp during Interval Cooling in Interval Total loss Corrected temp. 11 h. 07-Om. 20-85° C. 20-26 -005 -005 20^86 07-5 21-30 21-08 •009 -014 2131 08-0 21-60 21-45 •Oil •025 21 ^62 08-5 21-77 21-68 •012 •037 21-81 09-0 21-90 21-84 •013 •050 21-95 09-5 22-01 21-96 •014 •064 22-07 10-0 22-08 22-04 -014 •078 22-16 10-5 22-17 2212 -014 •092 22-26 11-0 22-20 22-18 -015 -107 22-31 11-5 22-23 22-22 •015 •122 22-35 12-0 22-26 22-24 -015 •137 22-40 12-5 22-28 22-27 -015 •152 22-43 13-0 22-30 22-29 -015 •167 22-47 13-5 22-31 22-30 •015 •182 22-49 14-0 22-32 22-32 •016 •198 22-52 14-5 22-32 22-32 •016 •214 22-53 150 22-31 22-32 •016 •230 22-54 Third Period.* Temperature falling steadily. Temp. Average temp. Cooling in Interval Total loss Corrected temp. 11 h. 15-5 m. 22-30° C. 22-31 -015 •245 2254 16-0 22-28 22-29 •015 •260 22-54 16-5 22-27 22-28 •015 •275 22-54 170 22-26 22-26 •015 •290 22-55 17-5 22-24 22-25 -015 •305 2254 180 22-22 22-23 -015 •320 22-54 18-5 22-20 22-21 •015 •335 22-54 190 22-20 22-20 -015 •350 22-55 19-5 22-19 22-20 •015 •365 22-56 20-0 22-17 22-18 •015 •380 22-54 20-5 2215 22-16 -015 •395 22-55 210 2213 22-14 •014 •409 22-54 21-5 2211 22-12 •014 •423 2264 Mean during last i^eriod Initial temperature ... 22-544 19-656 Riae of temperature * 8m Note at the end of the Section. 2-888 8. P. 130 HEAT XXV Calculat{o7i of Rate of Cooling during third period. 22-30 - ■ 22-19 = 11 22-28 - - 22-17 = 11 22-27 - 2215 = 12 22-26 - - 22-13 = 13 22-24 - - 22-11 Mean 13 120 Hence cooling in 4 minutes = 0° -120 )j » 2 5> = 0° -015 This cooling corresponds to a mean temperature of 22°*2 which is obtained with sufficient accuracy from the temperature observed at the middle of the third period. In an accurate determination of specific heats we are able to measure a rise in temperature of a few degrees to about one part in a thousand, although the observations have only been taken to the hundredth part of a degree. The increased ac- curacy is obtained by making use in the initial and final stage of a number of observations and taking the mean. In the present instance, the initial temperature of the india-rubber, owing to its treatment, must be doubtful to one degree, and the l£ist figure need not be taken into account in the final calculations, although it should be worked out as in the example given for the sake of practice. Knowing the masses of india-rubber and water, and the water equivalents of the thermometer and calorimeter, the specific heat of india-rubber can now be calculated. The final results are entered as follows : Weight of india-rubber = 1264 grams calorimeter = 28-14 „ „ „ and water = 155-9 Mass of water = 127-8 Calorimeter and thermometer equivalents = 3-2 Total water equivalent = 1310 Initial temperature of india-rubber = 100° C. Final = 22°-5 Fall in temperature „ „ = 77°-5 Rise of temperature of water = 2°-89 Hence specific heat of india-rubber = 0-387 XXV SPECIFIC HEAT OF INDIA-RUBBER 131 Draw curves (Fig. 50) shewing the observed changes of tem- perature and the temperature curve after correcting for cooling. jal -^^ ^ T^ :& 3E as ie i£ Tlm« -XL Fig. 60. The circumstances under which the loss of heat has been determined during the first and third periods are not strictly the same. At first the india-rubber was not immersed, and therefore the water equivalent was different, and also the level of the water in the calorimeter. This intnxiuces an error in the cooling correction which is very small when the change of temperature during the first period is small. Note. When unnecessary decimals are discarded, it is an elementary rule to increase the last remaining figure by one, when the first discarded figure is higher than five or five followed by other figures. But when only one figure is rejected, that figure being five, some doubt may arise, as to whether to leave the last figure as it stands or to increase it by one. We should commit a systematic error, if in the last column of the table on page 129, we were uniformly to take either the higher or the lower estimate. An excellent rule, adopted in the United States and deserving to come into general use, is always to leave the last figure an tvtn number. Thus 2535 and 30*65 should be shortened into 25*4 and 30*6. 9—2 SECTION XXVI. SPECIFIC HEAT OF QUARTZ. Apparatus required: Calorimeter with water jacket, thermometer graduated in tenths of degrees, piece of quartz, steam-jacket heater. If the specific heat of a substance is to be determined accurately, the method of heating adopted in the previous exercise should not be used, since it results in an uncertain amount of hot water being carried over with the heated body into the calorimeter. The body to be heated is suspended by a thread in an air chamber warmed by steam or hot water, and allowed to remain till the reading of a thermometer placed in I Fig. 51. XXVI SPECIFIC HEAT OF QUARTZ 133 the chamber has been constant sufficiently long to secure uniformity of temperature throughout the heated body. Fasten about 50 cms. of thread to the piece of quartz provided, and suspend the quartz in the centre of one of the steam-jacket heatere (Fig. 51). Measure the diameter of the thermometer, the length of its bulb and of that part of its stem which is immersed in the water of the calorimeter ; hence calculate the volume to about '5 cc. Assuming it to be all mercury, calculate the water equivalent. As the heat capacities of equal volumes of glass and mercury are not greatly dififerent, the equivalent of a thermometer may often be calculated in this way with sufficient accuracy. While the substance is heating fill the calorimeter about two-thirds full of water at the temperature of the room, the quantity of water used being found by weighing the calorimeter before and after filling. When the thermometer in the heater has been steady for 15 minutes, commence to take the observa- tions of temperature of the calorimeter for the " First period," as described in the previous section. At the end of this period, place the calorimeter and jacket under the heater and let down the quartz into the calorimeter as quickly as possible. Replace the calorimeter in its original position, take obser- vations during the " Second " and " Third " periods, and thence calculate the specific heat of quartz as in the previous section. Calculate the specific heat also by the following approximate method: determine the rates of change of temperature per interval of time during the first and third periods, and thence deduce the rate of loss at the temperature, which is the mean of the temperatures at the commencement of the second period, and of the maximum temperature attained. Multiply this rate of loss by the number of intervals between the commencement of the second period and the time at which the maximum temperature is attained, and add the product to the maximum temperature. This will, if the time is short, i.e. if the substance gives up its heat rapidly to the water of the calorimeter, be approximately the temperature which the 134 HEAT XXVI calorimeter would have attained if there had been no loss of heat. Record the calculation as follows : 10 Nov. 1898. Rate of cooling during first period = 0°*01 per half minute. Temp, on dropping quartz in ... =19°*72C. Maximum temp, attained ... = 22°"99 Interval between dropping quartz] _ « i ir • . and max. temp. ... ...J Rate of cooling during third period = 0°-025 per half min. Mean rate of cooling = 0°-l 7 5 per half min. .-. Loss of temp. ... = '0175 x 3 = 0°-052 Corrected maximum temp. ... = 23°04 Rise of temp. ... ... ... = 3°*32 Water heated + water equivalent =181 grams. .'. Heat absorbed by water ... =601 gram-degrees. Initial temp, of quartz ... ... = 99°'8 Final „ „ =23°-0 Decrease of temp. ... ... = 76°"8 Weight of quartz = 41*2 grams. . • . Specific heat of quartz = —^ — =-^ = 191. It is instructive to compare the results obtained by this approximate method of treating the cooling with the more accurate one given in the last section. It will be found that even in the case of a substance taking up the temperature as quickly as quartz the difference is appreciable. SECTION XXVII. LATENT HEAT OF WATER. Apparatus required : CaloHmeter, thermometer graduated in tent f IS of degrees, piece of ice. The latent heat of water, i.e. the heat absorbed when 1 gram of ice melts, may be determined by adding ice at 0°C. to sufficient water to melt it completely, and determining the decrease of temperature of the water. Weigh the calorimeter provided, place in it about 170 grams of water at about 18° C, and weigh again. Select a piece of ice weighing about 10 grams. Place in the water a thermometer gi-aduated to tenths of degrees, and observe the temperature of the water every half minute for 6 minutes (" First period ") as explained in Section XXV. Dry the surface of the ice thoroughly with blotting paper, and at the end of the next half minute drop it into the water of the calorimeter. By means of the stirrer keep the ice under water and the water stin*ed. Take half-minute observations of temperature till it reaches a minimum and then commences to rise — " Second period." Continue observations of temperature for 6 minutes after the rise has become uniform, these obser- vations constituting the ** Third period." From the observations taken in the first and third periods plot the curve from which the cooling at any temperature can be found, and apply the correction for cooling (or in this case heating) as described in Section XXV. Weigh the calorimeter to determine the amount of ice 136 HEAT XXVII which has been added, and from your observations determine the latent heat of water. Work out from elementary principles the formula applicable in this case and record as follows : 10 January, 1899. Initial temperature of calorimeter . , Final corrected „ „ Fall of temperature ... Weight of calorimeter and stirrer . . „ cal. + water „ water „ cal. + water + ice „ ice added Water equivalent of calorimeter „ „ thermometer .. Total,, „ = 16°-48C. = 12°-24 = 4°-24 = 59 '1 grams. = 2300 = 170-9 = 238 2 = 8-2 = 5-3 = '5 = 176-7 T ^ ^i. X 176-7 X 4-24 ,_^. ^„ , .-. Latent heat = ^^ 12*24 = 79-1. SECTION XXVIII. LATENT HEAT OF STEAM. Apparatus required ; Calorimeter, condenser, thermometer ^ flash, stand and burner. The latent heat of steam, i.e. the heat necessary to convert water at 100'^ C. into steam at the same temperature, may be determined by condensing a known weight of steam and observing the heat given up during the process to the liquid of the calorimeter, generally water, kept always below 100" C. The condensation takes place more regularly if the steam is allowed to condense in a separate vessel and not in the water of the calorimeter itself. f^ A-r-^ Fig. 52. Fig. 68. Weigh the calorimeter stirrer and condenser (Fig. 52) provided, first empty, then remove the condenser, weigh it to 138 HEAT XXVIII •01 gram, replace, fill the calorimeter to within about 2 cms. of the top with water at the temperature of the room, and again weigh. Arrange the vessel in which the steam is to be generated, the delivery tube and the calorimeter as shewn in Fig. 53. When this has been done, disconnect the tube from the condenser and place a burner under the boiler, regulating the height of the flame so that the water boils gently, the steam being allowed to escape from the end of the tube. Observe the temperature of the water in the calorimeter for 6 minutes — the " First period " of previous sections. Then replace the calorimeter, and insert the delivery tube into the head of the condenser tube so that the steam passes into the condenser and is condensed there. Keep the water well stirred, and observe the temperature every half minute, "Second period," till it has been raised about 10° C, then remove the delivery tube from the condenser. Continue observations of temperature till the change of temperature has been uniform for 6 minutes, " Third period." Weigh the calorimeter and contents. Determine the cooling during the first and third periods and by means of the values found correct the observed temperature throughout the experiment for cooling, and find the latent heat of steam at the boiling point corresponding to the atmospheric pressure at the time. Work out the necessary equation and record as follows : 7 December, 1899. Barometer 7 5 '3 cms. Boiling point of water = 100° - -37 X 7 99°-74 Weight of calorimeter, stirrer and condenser 91-82 grms. „ „ „ „ condenser and water ... 223-24 „ .-. weight of water 131-42 „ Water equivalent of calorimeter, stirrer, condenser and thermometer 8-8 „ . • . total water equivalent 140-2 „ XXVIII LATENT HEAT OF STEAM 139 Initial temperature ... ... ... 14°*76 Final temperature corrected according to method of Section XXIV 32°-12 Rise in temperature ... ... ... 17°*36 Final weight of water, calorimeter, etc. ... 22714 grms. Weight of steam condensed 3*90 grms. Latent heat of steam = — — ^^- 82*2 = 542 Experiments on latent heat of vaporisation are liable to a number of errors owing to the difficulty of taking account of the gain and loss of heat at the point where the steam is led into the apparatus. Hence the results obtained, when the experiments are made on a small scale are very uncertain. The above represents an average determination with the apparatus used. SECTION XXIX. HEAT OF SOLUTION OF A SALT. Apparatus required : Small calorimeter with suspending hook, larger calorimeters, thermometers, and salts. If p grams of a salt, the molecular weight of which is m, be dissolved in P grams of a solvent of molecular weight M, the solution formed has pjm gram molecules of the salt to PjM gram molecules of the solvent, or 1 gram molecule of the salt to every n = PmjpM gram molecule of the solvent. If the specific heat of the solution formed be c, and if during the process the temperature of the solution decreases from to to t, the quantity of heat absorbed by the solution of the salt is [{P^p)c-^w][t,-t], where w is the water equivalent of the calorimeter and thermo- meter. The quantity { {P+p)c + w]{t,-t] ^ P is the heat of solution of 1 gram of the salt, and the quantity is the heat of solution of 1 gram molecule of the salt, and is called the " molecular heat of solution." The molecular heat of solution of a salt is nearly constant for weak solutions, but diminishes as a rule as the strength of the solution increases. XXIX HEAT OF SOLUTION OF A SALT 141 Determine the molecular heats of solution of Sodium Chloride and of Ammonium Chloride in water by mixing 20 grams NaCl (m = 58'5) in 98'5 grams water (M— 18), n — 16 20 „ „ „ „ 147-7 „ „ „ 71 = 24 15 „ NH4Cl(m = 53-5)inl01 „ „ „ n = 20 li 151 71 = 30 The specific heats of the solutions may be taken as 84, '87, '87 and 89* respectively. Proceed as follows : Place the requisite quantity of water at about 18° '5 C. in a calorimeter surrounded by an air space and water jacket at the temperature of the room. Weigh the salt, put it into one of the small calorimeters, and suspend it by means of its hook in the water of the large calorimeter. Place a thermometer graduated to tenths of a degree in the water. After about 10 minutes take observa- tions of temperature for 6 minutes. If the change of tempera- ture is regular, unhook the small calorimeter and upset it in the water so that the salt and water come into contact with each other. Stir the mixture well and observe the temperature every half minute till the change has been regular for at least 6 minutes. From your observations determine the molecular heat of solution in each case, recording in the usual way and making the proper corrections for cooling. Tabulate your results as follows : Molecules H,0 ( t 8«lt p P to lott^ ••It to observed oorrected Q Q NaCl 20 98-5 16 18'-12 16*14 1405 11-4 660t )f 20 147-7 24 18-05 16°'48 16-38 13-4 770 NHCl 15 101 20 2r-32 13°-21 1301 60-5 3200} »> 15 151 30 2rio 14°-43 14-27 72-6 3900 « Von Buchka, TabelUn, pp. 275, 276. 1 H Berthelot, Therviochitnif, ii. p. 202. 1 ^ Ibid. p. 222. SECTION XXX. THE MECHANICAL EQUIVALENT OF HEAT OR SPECIFIC HEAT OF WATER IN WORK UNITS. Apparatus required: Pulujs friction cones with rotating pulley, jar of water, float, thermometer. When a gram degree of heat, i.e. the heat necessary to raise 1 gram of water at 15° C. to 16° C, is generated by the per- formance of mechanical work, the work done is called the mechanical equivalent of heat or the specific heat of water in work units (ergs per degree). To determine this quantity, the work may be done in a variety of ways; the one adopted in what follows depends on Fig. 54. one solid being made to slide along another against friction. In order that the sliding motion may be continuous the solids XXX MECHANICAL EQUIVALENT OF HEAT 143 are circular, one, a small hollow cone of steel, fits into another similar cone slightly larger. The lower outer cone is held in a frame which can be set in rapid rotation about a vertical axis coincident with that of the cone. The smaller cone is filled with mercury and is placed in the rotating cone but is pre- vented from rotating by a light wooden arm. To one end of this arm a thread is attached which passes over a pulley and carries a float placed in a jar of water. The moment of the couple which the tension in the thread exerts on the inner cone is equal and opposite to that which the rotating outer cone exerts on the inner cone. The work done by the frictional couple in any interval of time is equal to this moment multiplied by the angle through which the outer cone has in the interval been rotated with respect to the inner cone. To determine this angle the apparatus is provided with two dial wheels which register the number of revolutions of the outer cone. The angle of rotation is 27r times the number of revolutions. The tension in the thread is equal to the effective w^eight of the float which is numerically equal to ^r x volume of float pulled out of the water. Take the two cones out of the supporting frame, see that their surfaces are clean and weigh them together. Weigh the screws by which the wooden pointer is attached to the inner cone. Fill the inner cone to within 3 mms. of the top with clean mercury and weigh again to get the weight of mercury. Taking the specific heat of the steel of the cones to be 'llO, that of mercury to be '033 and that of brass '09, calculate the water equivalent of the cones and contents. Replace the outer cone in the supporting frame, taking care that it does not touch any of the metal of the frame. By means of the adjusting screws at the sides of the frame, centre the cone accurately so that it revolves about its own axis. Attach the wooden pointer to the inner cone and place the cone in the outer as in the figure. Adjust the position of the float cylinder and the length of the thread so that when the cone spindle is rotated at a convenient speed the thread and the wooden rod are perpendicular to each other. See that 144 HEAT XXX readings of the surface of the water can be taken on the graduated scale of the float, both when it is raised by the tension of the thread, and when the thread is quite slack. The difference between the two readings should be 5 to 10 cms. This can be secured by using a little petroleum as lubricant between the cones if the difference is too gi-eat, and a little vaseline if it is too small. Take the reading of the float when the tension in the thread is zero. Read the two dials attached to the rotating apparatus, and measure the length of the wooden rod from the centre of the cones to the point of attach- ment of the thread. Measure the diameter and length of the bulb of the thermo- meter and calculate its water equivalent. Hang the thermometer on the movable arm attached to the stand and lower the bulb into the inner cone till it is below the level of the mercury. Take observations of temperature every half minute for 3 minutes, then wait 3 minutes and take observations from 3 more minutes. At the end of this interval commence to rotate the hand wheel steadily, continuing to observe the temperature every half minute. At the middle of each half minute take a reading of the surface of the water on the float. Continue the rotation till the temperature has risen about 5°C. Then stop the rotation, read the temperature till the change has been regular for 3 minutes, wait 3 minutes, and read again for 3 minutes. Read the dials and the float. Remove the float from the water and measure its cross- section at 3 or 4 points, where the readings have been taken. Take the mean of these measurements. Take also the mean of the readings of the float in the raised position, and subtract from it the mean reading for no tension. The product of this difference into the mean cross-section of the float is the mean tension in gravitation measure. Multiply this by Stt times the number of revolutions and by the length of the wooden arm and by g, and the product is the work done during the rotation in ergs. Apply the correction for cooling to the temperature readings as in Section XXV. and determine the corrected rise of tem- perature. The product of this by the water equivalent of cones and contents is the heat generated in gram-degrees, i.e. in terms XXX MECHANICAL EQUIVALENT OF HEAT 145 of the unit of heat to which the specific heats used have been referred. The quotient of the number of ergs of work done by the number of gram-degrees of heat generated is the number of ergs work required to generate heat sufficient to raise 1 gram of water 1 ° C. Arrange your observations and results as follows : — 10 June, 1899. Weight of steel cones ., „ and mercury „ mercury Water equivalent of cones . . . 84'4 x „ „ mercury ... 149-2 x thermometer ... Total water equivalent of cones and contents Reading on stem of float without tension = Readings during rotation : — 19-70, 19-60, 19-55... &c., mean Mean rise of float ... Initial reading of counter ... Final „ ., No. of revolutions ... Angle turned through 27r x 1765 Length of arm of lever Cross-section of float stem = 102, 1-01, 1-00, -98, -99, mean .-. Work done = 257 x 13 x 1 x 11090 x 981 = 363 x 10' ergs. Temperature at end of first period 15'' 00 C. Mean temperature at end of third period) on^.?' after coiTecting for cooling J Rise of temperature ... ... ... ... 5°'75 Heat generated = 1493 x 5 75 = 858 gram degrees. 363 X 10' 858 ' = 84*4 grams. = 233-6 „ = 149-2 „ 119 = 9-76 033 = 4-97 ... = 10 ... = 14-93 = 65 2 cms. = 19-52 „ = 130 „ = 105 = 1870 = 1765 =11090 radian.^. = 257 cms. = 100 sq. cm. Mechanical equivalent = = 42 '3 X 10* ergs per degree. 8. p. 10 BOOK IV, SOUND. SECTION XXXI. FREQUENCY OF A TUNING FORK BY THE SYREN. Apparatus required : Tuning fork, singing flame, syren and blowing apparatus. To enable the comparison of a fork and a syren to be made more conveniently than it can be done directly, it is usual to tune a singing flame to the fork by adjusting the length and position of the resonating tube over the flame, and then to tune the syren to the flame. Calculate the length of an open pipe which will act as a resonator to the fork, the vibration frequency of which is supposed to be known roughly. Take a glass tube of rather less length, and roll a piece of paper round one end so that the effective length of the tube may be altered by sliding the paper tube along. Place the tube above a small gas flame produced by gas issuing from a minute hole at the end of a conical glass tube. Bring down the pipe on to the flame, and adjust till the flame " sings " ; then vary the position of the paper tube till the note emitted by the pipe produces no beats with the note of the fork (Fig. 55). XXXI FREQUENCY OF A TUNING FORK BY THE SYREN 147 Place the syren on the blowing apparatus, start the appa- ratus and increase the rate of blowing till the note emitted by the syren produces no beats with the pipe. Fig. 65. Maintain the rate of blowing, and at a given instant take a reading of the positions of the fingers on the dials indicating revolutions of the spindle of the syren. At the end of a minute again take readings. Subtract the readings to get the number of revolutions in the interval, count the number of holes in the disc of the syren, find the product and divide by 60, the result is the frequency of the note of the syren, and hence of the pipe, and fork. Tabulate your results as in previous exercises. 10—2 SECTION XXXIL VELOCITY OF SOUND IN AIR AND OTHER BODIES BY KUNDT'S METHOD. Apparatus required : Kundt's apparatus, rods and rubber. Kundt's apparatus consists of a glass tube of about 200 cms. length and 5 cms. diameter into one end of which a rod of wood, metal or glass projects (Fig. 56). ^r- Fig. 56. The rod is securely clamped to the table at its middle point, and can be set into vibration parallel to its length by stroking it, if wood with a piece of cloth on to which a little resin has been rubbed, if metal with a leather rubber similarly treated, or if glass with a damp cloth. The end which projects into the tube is provided with a cardboard disc which has a diameter a little less than that of the inside of the tube. The tube contains a little lycopodium powder or finely divided silica and the further end is stopped by a movable plug. It is important that both tube and powder should be quite dry. When the rod is set into logitudinal vibrations, the disc on the end in the tube sets the air in the tube in vibration and the powder is carried along with the air. If the tube is long enough, there are certain parts of it where the air is not in XXXII VELOCITY OF SOUND IN AIR AND OTHER BODIES 149 motion along the tube, and at these parts the lycopodium or silica remains in heaps undisturbed. These heaps therefore indicate the positions of the nodes of the vibrating column of air, and twice the distance between consecutive heaps is the wave length of the vibration in the gas witlj which the tube is filled — in this instance air. Since the rod is clamped at its centre, this point will be a node in the rod, and the two ends will be the centres of vibrating segments, so that the wave length of the vibration in the rod is twice the length of the rod. Since the frequencies of the rod and of the gas in the tube are identical, the ratio of the distance between the two heaps of lycopodium or silica to the length of the rod, is the ratio of the velocities of sound in the gas and in the material of the rod. Determine by this method the velocity of sound in brass and in glass, given the velocity of sound in air, and compare the velocities of sound in air and in coal gas. Clean and dry the tube provided, and scatter in it a small quantity of finely divided silica. Clamp the brass tube at the middle point to the support, and clamp the support to the table. Slide the tube over the rod and support it on blocks so that the cardboard disc at the end of the rod can move in the tube freely. Tap the tube sharply so that the silica collects in a line along the tube. Rotate the tube about its axis through a few degrees, so that the line of silica is not at the lowest point of the tube. Rub the rod lengthwise with a piece of leather on which resin has been rubbed, watching the line of silica during the process. If no motion is perceived, move the plug at the further end of the tube a centimetre and repeat. Continue till a position of the plug is found for which, when the the rod is rubbed, some of the silica is blown along the tube and falls to the lowest point in a number of heaps. Move the plug by millimetres at a time till this action appears most energetic, and when the heaps are distinct and nearly touch each other, count their number and observe on the scale under the tube, the p<:)sition8 of three consecutive spaces between the heaps at each end of the tube. Take the mean of the first three and the mean of the last three, subtract and divide by 150 SOUND XXXII the number of heaps between the mean readings, i.e. find the distance between consecutive heaps. Observe the temperature of the air in the neighbourhood of the tube, and measure the length of the brass rod. Substitute for the brass rod one of glass, and repeat the experiment, then one of wood and again repeat. Record as follows : 21 January, 1897. Brass tube 129 "5 cms. long. Readings of centres of intervals between heaps of silica : 3000 cms. 91-50 cms. 42-30 103-80 5470 116-20 Means 42-33 10383 Difference = 61*50 cms. for 5 intervals. Distance of heaps apart = 12-30 cms. Temp, of air... ... =18"C. Velocity of sound in brass / velocity of sound in air = 129-5/12-3 = 10-53. Velocity of sound in air at 18° C. = 341 metres per second. .-. velocity of sound in brass = 3590 metres per second. Similarly for the other rods. By fitting corks to the end of the tube and filling it with some other gas, the velocity of sound in that gas may be compared with the velocity in air. SECTION XXXIII. STUDY OF VIBRATIONS OF TUNING FORKS BY MEANS OF LISSAJOUS' FIGURES. Apparatus required : Two large tuning forks, one the octave of the other, lens and drop of mercury. When a tuning fork is set into vibration by one of the prongs being displaced from its normal position with respect to the other, both prongs are set into oscillation by the elastic forces which resist deformation of the fork. These forces depend mainly on the cross section of the prongs near their roots, while the masses which most influence the movement are those near the ends of the prongs. The influence on the frequency of a fork of a small mass added to a prong near one end, may be readily studied by the help of '' Lissajous' figures." Arrange the two large tuning forks provided, so that the directions of vibration of the prongs are horizontal and vertical respectively. To the end of the upper prong of the fork vibrating vertically, attach a small lens of about 7 cms. focal length, and to the lower prong a small weight to counter- balance the lens. Place the second fork in such a position that the outer end of one prong is seen through the lens distinctly (Fig. 57). Rub a little grease on the end surface of the prong, and rub a drop of mercury into the grease with the finger. The drop will break up into a number of smaller drops which will adhere to the surface. Move the fork till one of the drops is seen distinctly through the lens. Strike 152 SOUND XXXIII the prongs of the two forks with a rubber stopper in the hole of which a short wooden rod has been inserted to serve as a handle. Fig. 57. On looking through the lens at the bead of mercury, a bright looped line is seen which changes its shape more or less quickly according to the relative frequencies of the two forks. The time which the curve takes tu go through a cycle of changes should be approximately noted. The adjustable weights on the prongs of one of the forks should then each be moved slightly, say towards the ends of the prongs, and the observation of the time of a cycle repeated. If it is greater than the previous time, the sliding weights should again be moved in the same direction, if less, in the opposite, till the time the curve takes to go through its changes of form is too long for the vibrations to be maintained throughout it. The two forks may then be considered in unison. Measure the distances of the weights from the ends of the prongs, move them both 1 mm. and observe the effect on the time the curve takes to go through its changes of form. Repeat the displacement of the weights till as many observations as possible have been taken on each side of the position for unison. XXXIII STUDY OF VIBRATIONS OF TUNING FORKS Tabulate the observations as follows : 153 Distance of weights from ends of prongs Displacement Time for gain of 1 vibration, seconds Vibrations gained per second Upper Lower 67 cms. 6-8 cms. cm. Unison 6-6 6-7 •1 10-4 •096 6-5 6-6 •2 8-4 •120 6-4 6-5 •3 6-8 •147 6-3 6-4 •4 50 •200 6-2 6-3 •5 4-0 •25 61 6-2 •6 3-48 •29 60 61 •7 2-92 •34 6-7 6-8 Unison 6-8 6-9 •1 100 •10 6-9' 70 •2 8-6 7-0 7-1 •3 7-2 71 (fee. 7-2 7-3 &c Draw a curve taking distance of the weights from the position of unison as abscissae and the vibrations gained per second as ordinates (Fig. 58). Fig. 68. BOOK V. LIGHT. SECTION XXXIV. MEASUREMENT OF ANGLES BY THE OPTICAL METHOD. If a body undergoes an angular displacement we may measure the displacement by reflecting a ray of light from a mirror attached to the body. If the plane of the mirror is parallel to the axis of rotation of the body, the reflected ray will turn through an angle which is double the angular dis- placement of the body. It is often found more convenient to measure the displacement of a ray of light reflected from such a mirror attached to the body than to measure directly the angle through which the mirror turns ; al- though theoretically as great an accuracy may be obtained by either method. In (Fig. 59) let XO be a ray of light incident on the mirror LM \ let ON be the normal and OK the reflected ray, K being the point at which the reflected ray cuts a line drawn through X at right angles to the incident ray. If a is the angle between the incident ray and the normal, the angle XOK = 2a and hence irZ=0Xtan2a (1). Fig. 59. XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 155 If now the mirror turns through an angle so that a + ^ becomes the angle between XO and the normal, we shall have A''Z = 0Xtan2(a + ^) (2), where K' is the point at which the reflected ray cuts ^^ in the new position of the mirror. From the two equations (2) and (1) we can determine 6 20 — arctan -^^y "" arctan j-y (3). If the line AB is divided into equal divisions K'X and KX can be read off directly and hence 6 calculated from (3) after OX has been measured. In order to make this method of measuring angles a practical one, we must of course deal with a beam of light and not with a single ray, and in that case the position of K will be well defined on the scale only when the beam comes to a focus at that point. In order to have the best definition the whole mirror should be filled with light, and hence the width of the beam at X must be twice the diameter of the mirror. Let in (Fig. 60) AB be the divided scale, L a lens with CTosH wires at F which can be illuminated from behind, then if the distance of the cross wires from the lens is properly adjusted, a real image will be formed at a point K of the scala On the other hand if K is a division of the scale suflficiently well illuminated, an image of A*" will appear at the point F. This image may be magnified by an eyepiece, and the observer looking through what is now a tele- scope with objective at L and eyepiece at S, will see the divisions of the scale move through the field of view as the mirror turns round. The fii-st method called "the objective method," in which the source of light is at /" and the motion Fig. 60. 156 LIGHT XXXIV of the image at K is observed, is now very commonly employed. The apparatus is compact and the readings can be taken more quickly though not so accurately as with the other method. The objective method will be generally employed when, as in the case of a Wheatstone bridge, we do not wish to measure the deflections of a galvanometer mirror accurately, but only to work with electrical adjustments such that there should be no deflections (null methods). It will also be employed when, as for instance in electrometer work, the unavoidable sources of error are so large that extreme accuracy in the reading of deflections would be waste of labour. The second or "subjective method" will generally be used when great accuracy is required ; but it is necessary, in order to secure this end, that the galvanometer mirrors should not be too small and that the whole scale of the dimensions should be increased. This of course might also be done with the objective method but other difficulties would then arise. The objective method requires that the scale should be properly shaded so that the spot of light may be easily seen ; the subjective method requires that the scale should be properly illuminated so that its divisions should clearly appear in the telescope. The former condition is more easily secured when the apparatus is of small, the latter when it is of large dimen- sions. The large dimensions have the additional advantage that the observer works sitting at some distance from the instrument and hence irregular disturbances due to his moving about are often avoided. In order to secure accuracy, it is important to adjust the position of the scale, and to apply a few corrections to the final result. We shall describe the adjustments for the case of in- struments in which the subjective method is used, as they are then of special importance and may more easily be carried out. The student will have no difficulty in applying what is said to be the objective method. We shall assume the mirror to turn round a vertical axis. The plan of the arrangement is that shewn in Fig. 60, but it is clear that the telescope must be placed so that its axis passes either above or below the scale. The scale, mirror and telescope will therefore all be at different XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 157 levels and the telescope must either point downwards or up- wards. This introduces no error in the result as long as OX in Fig. 59 is taken to be the horizontal projection of the optical axis of the telescope, as will be seen by considering that the image of each vertical scale division will also be vertical. If the telescope is therefore turned round a horizontal axis, the same division will alwax's remain against the cross wire, and though we may have to shift the scale and mirror in a vertical plane in order that the image may appear in the field of view, no correction need be applied for want of horizontality of the optic axis. If the diameter of the mirror is equal to half that of the telescope, we can only obtain the maximum amount of light if the optic axis cuts the centre of the mirror, and in any case it is best to secure this for reasons of symmetry. The adjustment may be made by focussing the telescope on the mirror instead of on the image of the scale, moving it if necessary until the cross wires cut the mirror symmetrically ; a little practice will however enable the observer to make the adjustment without altering the focus of the telescope, for its field of view is generally sufficiently large to see in addition to the scale a blurred image of the galvanometer parts immediately surrounding the mirror. If the clear image of the scale is seen in the centre of the field of view, the adjustment is sufficiently accurate. In the equations (1), (2) and (3) it is assumed that the scale stands at right angles to the horizontal projection of the optic axis, and this adjustment must be made to the necessary degree of accuracy. This may etisily be done, as the following investi- gation shews that the error intro- duced by a slight deviation from the correct position can readily be eliminated from the result. Let the projection of the optic axis cut the scale at X (Fig. 61) an<l the mirror at 0, also let A and B be points on the scale equidistant from X, then if the scale is properly adjusted OA = OB. The scale occupying the position Fig. 61. 158 LIGHT XXXIV A'R, let 7 be the angle BXB' and </> be the angle between OX and OB. The triangle OXB' gives B'X sin<^ sin <i> OX ~ . ( IT \ ~~ cos (7 — <^) or B'X=^OX -^-^A (4). cos 7 4- sin 7 tan <f> The angle 7 is supposed to be so small that its cube may be neglected compared to unity, and (f> being also small we shall neglect the higher powers of 7tan</>; the result will shew when these simplifications may safely be made. If then we put cos7=l— J7-, sin 7 = 7, equation (4) becomes B'X= , OXl^ncj > 0Ztan</)(l-7tan<^ + l72) 1+7 tan <p — J7^ ^ ^ ^ approximately. If the scale had been placed in its proper position the line OB' would have cut it in B so that BX = OX tan (/>. The error in the reading will therefore be OX tan </) (- 7 tan </) + ^7^). If the mirror is deflected through the same angle but to the other side we find similarly the error of reading OX tan (f) (7 tan <!> + ^y^). If then 7 is so small that 7^ may be neglected the readings of the scale will be as much too great on one side as they are too small on the other, and hence if the mean of the two deflections is taken the error will only be dependent on the square of the angle through which the scale is turned. Our adjustments must be made therefore with suflScient accuracy to allow ^y'^ to be negligible compared to unity. If the scale is one metre long and divided into millimetres, we may estimate the tenth part of a division and therefore we may if necessary aim at an accuracy of one part in ten thousand. The angle 7 for which ^^ is '0001, is about 45', so that the inclination of the scale should not exceed this value. The XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 159 difference in the deflection right and left are according to the above formulae 2OZ7 tan» (t> = 2X^7 tan <^. In actual experiments tan (f> is not likely to be more than J, and if 7 is 01, the differences in the reading would be XB/200, If the scale is a metre scale so that the greatest value of XB is 50 centimetres the difference in the reading of equal deflections on the two sides will for the greatest deflection which can be read by the instrument be 25 mms. If then the mirror is deflected so that the end of the scale appears against the cross wire, and two ecjual deflections on opposite sides do not give a difference in the reading of more than two and a half scale divisions or generally *0025 of the total length of the scale, the adjustment is sufficient even if the greatest possible accuracy is required. As it is difficult to produce with certainty equal deflections, it is advisable to test the position of the scale by some direct measurement. Fig. 61 shews that OA' XA' OA' ^ hence OB' XB' ' OA'-OB XA'-XB' OB' XBf Substituting XA' - XB' = 2X^7 tan we find to the required accuracy OA' -0B' = yAB, which determines the greatest allowable difference between the lengths OA' and OB'. By means of a string or long rod it is not difficult to secure that the two ends of the scale shall be at the same distance from the centre of the mirror to within the hundredth part of the length of the scale, which will secure a maximum value of 01 for 7. If the mirror is enclosed in a case with a glass window it will be sufficient to measure to the centre of the window. To obtain the angle of deflection we use the formula (1) and deduce 2a = arctan ^ y . 160 LIGHT XXXIV For small angles it is often more convenient to calculate KX arctan yyy than to look it up in a table of tangents. Using the series arctan </> = </) — J <^^ + -J</>'' — . . . &c., we obtain by substitution, writing x for KX and d for OX, ''~2U 3rf»'*'5rf» '"j'UV Sd'~^5d'~ '")' The convenience of the formula consists in the fact that the terms after the first may either be neglected or determined approximately and that if only relative values are required we only need to calculate the bracket. It is often necessary to find tan a, sin a, or sin ^ , instead of a and the following approximate formulae may then be employed : tan<^ = ^[l-gy 2 U \ 2 VW J • sm These formulae will be sufficient for nearly all purposes, but in case the third term of the series has to be taken into account, the first formula becomes If it is not necessary to obtain the values of tan (f>, but only values which are proportional to them, it is easier to calculate the value of 2c^ tan </>, 2d sin <\> or ^d sin - . Thus for instance, if x is 392*3 scale divisions and if the distance d of the scale from the mirror is 2500 scale divisions the calculation might be made and tabulated as follows : XXXIV MEASUREMENT OF ANGLES BY THE OPTICAL METHOD 161 x= 392-3 ^ = •07846. (^J=00616. -.(^y= - 2-41 (^/= 00048. .{£)'= +^^ 2d tan 6 = 3901 In the above investigation it has been assumed that the normal to the mirror is at right angles to the axis of rotation, and this is sometimes difficult to secure with great accuracy. Calculation shews however that when the deviation of the normal does not exceed one degree the results are sufficiently accurate for nearly all purposes, provided that the distance d is measured from the centre of the mirror along the line which stands at right angles to the plane of the scale, to which the axis of rotation is supposed to be parallel. If as is commonly the case the axis of rotation is vertical, d should be the horizontal distance between the centre of the mirror and the plane of the scale. Should it be necessary in exceptional cases to take account of the inclination of the mirror, this can be done by adding xa (y — a) to the observed deflection x, where a and 7 are the angles which a plane drawn at right angles to the axis of rota- tion forms with the normal to the mirror and the optical axis of the observing telescope respectively. The angles a and 7 must of course be measured towards the same side of the plane *. * See F. Kohlrausch, Wiedemann Annalen xxxi. (1887). 8. P. 11 SECTION XXXV. THE SEXTANT. Apparatus required : Sextant, fixed marks. The sextant consists of a graduated arc of a circle AB (Fig. 62), of about 60°, and a movable arm /, which turns about Fig. 62. the centre of the arc, and is fitted with a clamp, a tangent screw, and a vernier by means of which its position on the graduated arc can be accurately determined. IG is a plane mirror attached to the arm I, and is called the Index Glass. HG is a second mirror fixed to the frame, and known as the Horizon Glass. Its upper half is left un- silvered. XXXV THE SEXTANT 163 T is a small telescope directed towards the mirror HG and placed parallel to the plane of the arc. By means of a screw at the back of the instrument, the telescope can be moved at right angles to this plane so as to vary the proportion of light received from the silvered and unsilvered portions of the horizon glass. The horizon glass is so placed that a ray of light passing from the centre of the index glass to the centre of the horizon glass, is after reflection directed along the axis of the telescope (Fig. 63). Let such a ray come originally from an object Q, and let another ray coming from a second Fig. 63. object P, pass through the unsilvered part of the horizon glass and proceed in the same direction. The two objects when viewed through the telescope are then seen to coincide, one being viewed direct and the other after reflection from the two mirrors, and the angle between the two mirrors, when this is the case, is half the angle between the rays from the objects. The graduations on the arc are numbered to read double their real value ; hence the reading on the arc gives directly the angle between two lines one of which is drawn from the centre of the index glass to Q and the other fiom the centre of the horizon glass to P. If the distance of the objects is sufficiently great this will be sensibly the same as the angular distance between P and Q at the observer's eye. 11—2 164 LIGHT XXXV Adjustments. 1. The plane of the index glass IG should be at right angles to that of the graduated arc. Verify that this adjustment is correct by setting the index at about 100°, placing the eye near the index glass, and looking obliquely at the glass so as to see at the same time part of the arc direct, and also its reflection in the glass. If the two appear to be in the same plane, the adjustment is correct, and the adjusting screws of the index glass need not be altered. 2. The plane of the horizon glass HG should be at right angles to that of the arc. Hold the instrument so as to view directly with the telescope some small distant object. On turning the index arm round, an image of the object, formed by reflection at the two glasses, will cross the field. If the two glasses are accurately parallel, this image can be made to coincide exactly with the object seen direct. If the plane of the horizon glass is not at right angles to that of the arc, so that the two mirrors are not parallel, the image will appear to pass on one side or other of the object. By adjusting the top screw at the back of the horizon glass, the image seen after two reflections, and the object seen directly, can be made to coincide exactly in one position of the index arm. When this is the case the two mirrors are parallel, and the horizon glass is at right angles to the plane of the arc. 3. The axis of the telescope should be parallel to the plane of the graduated arc. To test whether this condition is fulfilled with sufficient accuracy, place two small sights of exactly equal height on the divided circle. These sights are conveniently made by bending strips of sheet brass into two parts at right angles to each other, so that when resting on the sextant their upper edges are horizontal and at the same level as the axis of the telescope, the sextant being placed on a horizontal table. If the sights are placed as far apart from each other as possible and in suitable positions, the same distant object may either be viewed through the telescope or by the naked eye looking XXXV THE SEXTANT 165 over the sights. A point on the distant object which may be brought into contact with the upper edges of the sights when looked at by the naked eye, should then appear in the centre of the field of view. For accurate work with the sextant the ad- justment of the telescope should be con*ect to within 10 minutes of arc or '003 in angular measure. Hence if the sights are placed at a distance of 15 cms. from each other, they should be carefully constructed so that the two upper edges are hori- zontal and within half a millimetre of the same height above the divided circle. If the adjustment is found wrong it may be corrected in well-made sextants by two small screws in the frame which carries the telescope. Determination of the Index Correction. When the two mirrors are parallel, it may be found that the Vernier index does not read exactly zero. The reading is termed the index error, and the index correction is the index error with the sign reversed. To determine the index correction, direct the telescope to a distant object and turn the index arm till the two images of the object appear in the field of view. Then clamp the index arm, and bring the images into coincidence by means of the tangent screw. Read the Vernier, and notice whether the reading is positive or negative. The reading of the Vernier with the sign reversed is the index coiToction. Exercise. Standing on the given spot, determine the angle subtended by the two given points. Hold the sextant in the right hand, with the arc downwards. Look through the telescope at the lower of the two objects, holding the sextant so that tlie plane of the arc passes through both objects. Move the index arm along the arc till both objects appear at the .same time in the field of view, then clamp the arm by means of the screw behind the scale, and bring the two images into coincidence by means of the tangent screw. 166 LIGHT XXXV Read the position of the index, repeat the observation, and record as follows. 10 May, 1899. The reading when the two images of the same distant object coincided was found to be as a mean of 3 observations — 2' 12". Readings when the images of the two given points coin- cided : 11° 30' 20" 30' 30" 30' 00" Mean IV 30' 17' Index Correction + 2' 12' . •. Angle subtended by the given points] ^ , „ at the point of observation J It will often be found impossible to find suitable objects at a great distance. If the distance is less than 1000 yards, and the measurements of different observers are to be compared with each other, care must be taken that the observer does not change his position during the observations and that the position is well marked. Students will get a good idea of the delicacy of the observations by observing two objects at a distance of 1000 yards and at an angle of not less than 10° from each other; receding from or approaching the objects by a yard ought to make an appreciable difference in the coincidence of the images. It would be difficult to name a more useful or instructive instrument than the sextant, and, if time allows, students should obtain a little practice in the determination of latitude and local time by means of it. The study of errors introduced by imperfections in the adjustment of the different parts, will form an excellent foundation for the study of other optical instru- ments. Consult: Chauvenet, Spherical and Practical Astronomy, vol. II. SECTION XXXVI. CURVATURES AND POWERS OF LENSES. Apparatus required : T^uo large lenses, one concave, one cojive./', one suiall convex lens, spherometer, scale, plane mirror, and simple optic bench. The radius of curvature of a spherical surface may be found by means of a spherometer, or by using the surface as a mirror, and determining the positions of conjugate points. Both methods will be used in what follows. Place the spherometer provided on a plane surface, e.g., a sheet of glass, and turn the milled head till the point of the screw just touches the surface. The exact point may be known either by the slight increase of the resistance to rotation of the screw, or by the rocking of the spherometer on the surface as soon as the point of contact is passed. Take three readings of the zero of the spherometer in this way, and then place it on the surface the curvature of which is required, in this case a lens. Again turn the screw till the point just touches the surface, making the adjustment three times and taking a reading each time. The pitch of the screw is marked on the spherometer. In order to determine the radius of curvature of the surface from these observations, we require to know the distances between the feet of the sphero- meter. To measure these distances Fig. 64. place the spherometer on a sheet of paper, and press it down gently, so that the positions of the feet are indicated by slight 168 LIGHT XXXVI depressions in the paper. If these points be A^, A^, A3 (Fig. 64), measure by means of a glass scale laid on the paper with the scale downwards, or by means of compasses, the distances -42^3, A3A1, A^A^, call them a, b, c. The radius of the circle CA, circumscribing the triangle ^1^2-^2 will be given by abc r — where 4VS(,S- a){S- ■b)(S- ^y 8 = "- 2 c •educes to r = a V3- In practice a, b, c will not be exactly equal but nearly so and in that case we may generally substitute for a the average value of the distance in the last equation. In order to see within what limits of accuracy this is allow- able we write s for the arithmetical mean of a, b, c and put a = s + a, b = s + ^, = 5 + 7, and by addition a + 6 + c = 35+a + /3 + 7, or since by definition 3s = a + 6 + c, it follows that a + /3 + 7 = 0. We may now put \/35 (s - 2a) (s - W) {s - 27) * If we expand the products we find that there is no term containing the first power of a, ^, 7 because these quantities only occur in the combination ot + y8 + 7, which is zero. If a, /9, 7 are considered as small quantities the cubes of which can be neglected and if we write «2 = a/3 + /S7 + 07 = -i(a^ + ;^^ + 7^), it is easily found that XXXVI CURVATURES AND POWERS OF LENSES 169 As an accuracy of one part in 500 will probably not be reached in this exercise unless the mean of a large number of determinations are taken, we may write r = whenever - is less than s 25 V3' If the second term cannot be neglected it is easily computed with sufficient accuracy. Now consider a section of the surface by a plane through AC and the centre of curvature of the surface 0, i.e., the circle ANDN' (Fig. 65). The plane through the points ^1^0-^3 is represented by the line AGD. Assuming that the spherometer is properly constructed so that when the in- strument is placed on a horizontal surface the point of the screw touches the surface at the centre of the circle circumscribing AxA<iAi, the distance {d) through which the screw has been moved will be CN when the instrument is placed on the spherical surface. Now GN.CN'^CA^ and CN' :=2R-CN where R^OA. Fig. 65. Hence or d{2R-d) = r\ J, 1 r' + d^ Calculate R from this equation and determine similarly the radius of curvature of the other surface of the lens. If the point of the screw instead of touching C passes through a point at a distance h from it, the error in the measured distance will be — h*/'2R, and the fractional error of the radius of curvature measured will be It^/r^. Verify by the following optical method the values found : I. For a Concave surface. Place the screen provided with the hole and cross wires at one end of the optic bench with the white side towards the centre of the bench. Place a bright Hame behind the screen. 170 LIGHT XXXVI Place the lens on the bench with its axis parallel to the axis of the bench, and the surface the curvature of which is to be measured towards the screen. Find the position of the surface when the image of the cross wires on the screen pro- duced by reflection at the surface is most distinct. The cross wires are then at the centre of curvature of the surface. II. If the surface is convex the radius of curvature may be determined optically by the following method : — C (Fig. QQ) is the lens the radii of curvature of which we have to determine. B is an auxiliary lens provided with a stop Fig. 66. of small diameter. A is the screen in the centre of which is a small hole with a cross wire, which is illuminated as shewn. Keeping the distance AB, which must be greater than the focal length of B, constant, we can find a position of C such that the rays of light, after passing through B, strike the surface of C perpendicularly, and returning along the same path, form an image of the cross wire on A. Take the reading of the position of G, then remove G, and place a screen D in such a position that a distinct image of the cross wire is obtained on it, the lens B being kept in the same position. The distance from the lens to D is then the radius of curvature of the face nearer the cross wire. By turning the lens C round, the radius of curvature of the other face is similarly obtained. The apparatus used is shewn in Fig. 67. Fig. 67. XXXVI CURVATURES AND POWERS OF LENSES 17 1 Deterraine approximately the focal length of the lens, if convex, by placing a screen behind it at such a distance that an inverted image of some distant object is formed on the screen. The distance of the screen from the nearest point on the surface of the leus is the focal length approximately. To determine it more accurately, if the lens is a converging one, mount it with a plane mirror immediately behind it. Place the mounted lens and the mirror on the optic bench. Behind the hole and cross wire place a luminous burner, and find the position of the lens and mirror for which a distinct image of the cross wire is projected on to the screen, in the same plane and as nearly as possible coincident with the cross wires themselves (Fig. 68). The light from F in the direction FA is refracted by the lens and proceeds after refraction in the direction AD. li AD is perpendicular to the surface of the mirror, the reflected ray traverses the same path as the incident, and is, therefore, brought to a focus again at F. In order to determine the distance of the surface of the lens from the screen read the position of some part of the lens-stand on the scale of the optic bench used. Move the lens towards the screen till it touches one end of a rod of known length the other end of which touches the cross wire. Again read the position of the lens-stand. The sum of the length of the rod and the distance through which the stand has been moved is the distance of the surface of the lens from the cross wire. The focal length may be found from this by adding to it J the thick- ness of lens. If the lens is a divergent one, place between it and the cross wires a convergent lens. Place behind the two and close to 172 LIGHT XXXVI the divergent lens, a plane mirror (Fig. 69). Move the diver- gent lens and plane mirror till a distinct image of the cross wires is formed on the screen close to the wires. Note the position of the lens, remove it and place behind the converging lens a screen in such a position that an image of the cross wires -^*-E- Fig. 69. is formed on it. The distance from the screen to the position previously occupied by the lens is the focal length required. The focal length of a lens is connected with the radii of curvature of its surfaces and with the index of refraction /i of its material by the equation 7=('*-^>(i+ii where R^ and i^ are taken positive for convex and negative for concave surfaces. From this equation we get flKR.'^R)' Calculate /x from the observations, and tabulate as follows : 10 February, 1899. Lens No. 2 double convex. a = — '03 cms. t- — '05 ^ = -•14 „ ^V«^ = -0014 7 = + -!'^ .. „ and r = 3*560 „ di by spherometer = '1564 cm. a = 6-14 b = 6-03 c = 6-34 5 = 6167 cms. Hence By reflection Hence 0/2 „ f (measured) fj, (calculated) = -1567 „ = 40*60 cms. = 40-52 „ = 39-9 „ = 40-0 „ = 40-5 „ = 1-50 SECTION XXXVII. DETERMINATION OF THE INDEX OF REFRACTION OF A LIQUID BY TOTAL REFLECTION. Apparatus required : Horizontal graduated scale with upright and dit, ebonite block, and glass cube. When a ray of light traversing an optically dense medium impinges on the surface of separation of that medium from a rarer medium, making an angle with the normal at the point of incidence greater than the " Critical Angle," the ray is totally reflected, no part of it entering the rarer medium. The least value of the angle of incidence for which total reflection can take place, i.e., the Critical Angle, is, if N is the index of refraction of the denser, n that of the rarer medium, given by the equation sin 6 = njN. A glass cube of about 4 cms. edge is provided. On one face a line parallel to and about a cm. from an edge has been drawn with a diamond. Put a few drops of water on the ebonite block provided, and place on it the cube with the face on which the line is drawn vertical, the line itself horizontal, and a cm. above the ebonite. A film of water will be formed in immediate contact with the lower face of the cube. Now place the ebonite and cube on the scale and about 30 cms. behind the slit. Place the scale horizontally on a table in front of a window through which the sky can be seen. Look through the slit "at the lower surface of the cube, and notice that on moving the cube and block towards the slit, the appearance of this surface changes from bright to 174 LIGHT XXXVII dull, and at the point of change a coloured band extends across the surface from left to right. Adjust the distance of the cube Fig. 70. from the slit till that part of the coloured band where violet shades off into green coincides with the diamond scratch on the face of the cube. Measure h^ the height of the slit, and h^ the height of the line on the cube, above the surface of the scale, and d the horizontal distance between the face of the cube and a per- pendicular let fall on to the scale from the slit. Fig. 71. XXXVII TOTAL INTERNAL REFLECTION 175 Then if i = angle of emergence of the ray from the cube, (Fig. 71), tan I — -1 — . If r = the angle of incidence of the internal ray at C, and N = the index of refraction of the cube, then sin 1 = iV sin ?*. Since total reflection just begins at B, i.e., the angle of incidence at 5 is the critical angle, 71 = N cos 7', where ?i = the index of refraction of the liquid at B. Squaring and adding the last two equations we get ?i2 + sin^ i = N^ or n = ViV^ — sin' i. From which n may be calculated if N is known and i is found by measurement of /i,, ho, and d. Determine by this method the value of N for the cube, taking the index n for water to be 1'333. Having found N find n for ethyl alcohol. Arrange your observations as follows : — 25 January, 1893. Water film h^ = 21*5 cms. d = 2l'S cms. A,= 31 „ 18*4 tan i = 2^3 =862 .-.1 = 40" 45', sin I =653 N' = (1-333)' + (653)2 = 22060 .•.iV^= 1-485 Alcohol film A, = 21*5 cms. ^= 31 „ ^--• = 2l| = -^21 1 = 35° 48', sin I =586 7i» = (1-485 )«-(-586)« .-. 7i = 1-365. d = 25*5 cms. SECTION XXXVIII. MAGNIFYING POWERS OF INSTRUMENTS. Apparatus required : Telescope, microscope, millimetre scales. When an object is viewed through an optical instrument, such as a telescope or microscope, the apparent size of the image is in general larger than that of the object as seen direct, and it is the purpose of this exercise to determine the amount of magnification. The magnifying power of a telescope is defined as the ratio of the apparent size of the object as seen through the telescope to the apparent size of the object as seen direct. To determine this ratio the telescope is directed towards some surface on which is a regular well-marked pattern of some kind, such as the dividing lines in a brick wall, and the pattern looked at with one eye through the telescope, while with the other eye the pattern is seen direct. Count how^ many times some convenient length in the magnified pattern contains the same length of the pattern seen direct. This number is the magnifying power of the telescope as used. Alter the distance of the telescope from the pattern, and again find the magnifying power, which should be the same as before if the distances are great compared to the focal length of the object lens of the telescope. The magnifying power of a microscope is defined as the ratio of the apparent size of an object as seen through the XXXVIII MAGNIFYING POWERS OF INSTRUMENTS 177 instrument, to the apparent size of the object when placed at the nearest distance of distinct vision and viewed direct. Place a millimetre scale on the stage of the microscope, and focus it distinctly. Hold a paper scale graduated in millimetres about 20 cms. below the eye and parallel to the scale on the stage, move it up and down till it is as near the eye as is con- sistent with distinct vision. Use one eye to view the scale on the stage through the microscope, the other to view the paper scale direct, and note how many divisions of the paper scale correspond to one of the stage scale. This is the magnifying power required. Describe your experiment in your note-book and enter the results as in the previous exercises. 8. p. 12 SECTION XXXIX. ADJUSTMENTS AND USE OF THE SPECTROSCOPE. Apparatus required: Spectroscope, platinum wires for heads, salts, and crayons. Vision through a prism. Let a luminous point S (Fig. 72) send out a pencil of homogeneous light, the rays of which, after re- fraction through the prism, seem to diverge approximately from a point ;S'. An eye at E will therefore see the luminous point in a displaced Fig. 72. position, the amount of displacement depending on the refractive index of the prism for the rays. If the light sent out by the luminous point is not homogeneous, but is of two kinds having different refractive indices, there will be two separate images side by side, which will be coloured differently if the difference in the refractive index is sufficiently great. If, finally, the luminous point sends out white light, there will be an infinite number of images shewing a succession of all the colours of the spectrum from red to violet. Seal about four or five centimetres of platinum wire of *2 mm. diameter into a piece of glass tubing, and bend the end of the wire into the shape of a loop thus : Fig. 73. XXXIX SPECTROSCOPE ADJUSTMENTS 179 Wet the loop slightly, dip it into a mixture of common salt and borax in equal proportions, and heat carefully in the edge of the flame of a Bunsen burner, until the salt fuses and forms a transparent bead. If the loop is not completely filled by the bead, place some more of the substances on it and repeat the process of fusion. Then fix the glass tube to the stand pro- vided, and place it so that the bead touches the flame and colours it. The platinum wire should dip slightly downwards (Fig. 73), otherwise the bead has a tendency to run back along it. Hold the loose prism provided in front of your eye, and turn your head until you see the displaced image of the flame. Next take a second piece of wire, and having prepared a bead of a lithium salt, place it in the flame simultaneously with the sodium bead. Make a sketch in your note-book of what you see. What conclusions do you draw respecting the light sent out by a flame containing both sodium and lithium ? Look now at a luminous flame, again sketch the appearance in your note-book, and explain it. Cut a slit not broader than a millimetre in a piece of card- board. Hold the cardboard in your left hand so that the slit is in front of a Bunsen burner containing both a lithium and sodium salt, hold the prism in the right hand in front of your eye and look at the image of the slit. Sketch the appearance in your note -book. Such a combination of slit and prism forms the simplest kind of spectroscope, and is often useful for the rapid examina- tion of light sent out by a flame, or an electric discharge. In practice the virtual image of the slit is magnified by being looked at through a telescope. But this cannot be done with advantage without some further change in the optical Fig. 74. arrangement, to obviate the so-called " aberration " of the rays. For if rays of light diverging from a point are traced through a 12—2 180 LIGHT XXXIX prism, it is found that after emergence they do not accurately diverge from a point, but the section of the pencil will have the appearance of Fig. 74. This does not sensibly affect the sharpness of the image when looked at with the naked eye, or even under small enlargements such as are used in pocket spectroscopes, but when higher power is required the aberra- tion must be obviated. This may be done by the introduction of a lens placed in such a position between the slit and the prism, that the beam of light coming from any point of the slit becomes parallel before it falls on the prism. As a parallel pencil of light will remain parallel after refraction at any number of plane surfaces, there is no aberration in this case. We thus arrive at the arrangement shewn in Fig. 75. Fig. 75. A tube carries at one end the slit S and at the other the lens L, the slit being in the focal plane of the lens. This tube is called a "collimator." The pencils of light rendered parallel by L fall on the prism P, and after passing through it are received by a telescope T. The eye, on looking through the telescope, will see an image of the slit, which is displaced by the prism, but is sharp provided the slit is illuminated by homogeneous light. If, on the other hand, the light falling on the slit consists of groups of waves differing in wave-length, each group will give a separate image of the slit. If further the wave-lengths, and therefore the refrangibilities, vary continuously over a certain range, the images of the slit will lie side by side or even overlap, so that a continuous band of light will be seen. We call the appearance presented when the light of a luminous body is examined by means of a spectroscope the "spectrum" of the body. We say that the body has a XXXIX SPECTROSCOPE ADJUSTMENTS 181 "contiDuous spectrum" when the band of coloured light is continuous. We say, on the other hand, that the body has a "line spectrum," if a number of separate coloured lines are seen, which students must be careful to remember are only images of the slit. If the slit is curved the lines will be curved, if the slit is broad the lines appear broad, and nan-ow- ing the slit will narrow the lines down to a certain limit, which depends on the diameter of the object-glass used. A "band spectrum " is a spectrum consisting of bands which are broad even with a narrow slit. These bands are often sharp on one side and fade away gradually on the other. As the spectrum of a body, whether it is a line or band spectrum, is found to be characteristic of the body, it is neces- sary to determine the positions of the lines and bands on some convenient scale. The arrangement generally adopted in one-prism spectro- scopes is shewn in Fig. 76. A small tube MQ has at one end a scale of fine equi- distant lines Q, at the other end a lens M, the scale being at the principal focal plane of the lens. The tube is placed so that the light rendered parallel by the lens M, is re- flected at the surface of the prism into the telescope T. When the scale is illuminated by a small gas flame, the observer sees not only the spectrum of the body, but superposed on it the image of the divided scale, and he can ^*^' ^^' read off the position of each separate line on this scale. Fig. 77 shews a spectroscope consisting of collimator, prism, telescope, and scale- tube. The collimator is provided with a projecting metal sheet, and the scale-tube with a projecting metal cylinder, to prevent the flames used being brought so near as to injure the instrument. Adjustments of the Spectroscope : It has been shewn above that the light leaving the collimator should consist of parallel rays. To secure this, if the distance between the slit and collimator lens can be altered, the telescope and collimator are adjusted as follows : — 182 LIGHT XXXIX The telescope is taken off the spectroscope stand, and a distant object is observed through it. If the object is seen Fig. 77. clearly, the telescope is in adjustment for parallel light. If not it should be adjusted and then replaced. Illuminate the slit of the collimator by a sodium flame, and move the collimator in and out till the image of the slit seen in the telescope is sharp and distinct. Place a small luminous burner in front of, but not too near to, the end of the scale-tube, and adjust the scale- tube till both the scale and the sodium line are. seen sharp at the same time. Move the scale-tube at the same time until the sodium line stands at a convenient number of the scale*. The best test of good adjustment is the absence of parallax; that is to say when the eye is moved a little to the left or right the sodium line should not change its position relative to the scale. Accurate measurements with the spectroscope are impos- sible when parallax exists to an appreciable degree. If the position of the slit of the collimator cannot be changed, the telescope must be adjusted by illuminating the slit with a sodium flame, and focussing the telescope so that the yellow line appears sharp. The scale is then focussed as described. Before proceeding with any further measurements, consult one of the Demonstrators to see that your adjustment is correct. With a narrow slit the sodium line should just appear double. As the question of brightness of spectra is often an im- * As the scale divisions may differ in different instruments, a label is attached to each spectroscope, on which the number with which the sodium line should be made to coincide is indicated. XXXIX SPECTROSCOPE ADJUSTMENTS 183 portant one, it will be useful to remember the following facts. Let SL (Fig. 78) be a horizontal section of the collimator, S being the slit and LL the lens. Let AB be a flame sending out light the spectrum of which is to be examined. The light from the part of the flame near A will illuminate the part of the collimator lens near Z, the rays of light going from A through the slit to Z, similarly the light from the part of the flame near B will illuminate the part of the lens near L. The whole collimator lens is thus filled with light. When the whole collimator levs is filled with light maximum brightness of the spectrum is secured. Fig. 78. If the burner is placed at ^^^j close to the slit, it is only the light from the central parts of the flame which reaches the collimator lens : the rest of the light falls on the inside walls of the tube, which are blackened to prevent reflections. In this case, therefore, although the flame is nearer, a smaller part of the flame illuminates the collimator lens, so that nothing is gained in brilliancy, the brightness of the spectrum being exactly the same as before. If, on the other hand, the flame is too far away at A^Bo, the outer portions of the lens are not illuminated at all ; in that case the brightness of the spectrum is reduced. When the collimator lens is not filled with light, as in this last case, beside loss of light there is the disadvantage of worse definition, so that in spectroscopic work it is generally of importance that the whole collimator lens should be illumi- nated, but when that is secured no further increase in brightness is possible. Where the source of light is very small, as in the case of an electric spark, with a narrow slit the collimator lens would not be filled with light unless the source were brought inconveni- ently near the slit. A lens P (Fig. 79), should therefore be placed in such a position between the spark E and the slit 8 that the rays traverse the paths shewn, and an image of the 184 LIGHT XXXIX spark is formed on the slit. Here, again, it does not matter how the distances of lens and spark from the slit are adjusted, nor how bright is the image of the spark seen on the slit plate. The filling of the collimator lens with light is the sole circum- stance on which the brightness of the spectrum depends. We Fig. 79. may even move the lens so that the image of the slit is out of focus without producing great loss of light. The use of the lens has the further advantage of enabling us to examine the spectra of different parts of the spark separately. Mapping Spectra : When the spectroscope is in adjust- ment and the sodium line at the proper scale division, place a luminous burner behind the slit of the collimator so as to produce a continuous spectrum. Draw the spectrum which you see, representing as nearly as possible with the crayons provided, the colours and shades and their extent on the spectroscope scale. Use for this purpose a sheet of paper on which scales with the distance between the scale divisions approximately equal to one millimetre are printed. Next prepare six pieces of platinum wire sealed into glass tubes, as described at the beginning of the section. The platinum should not be thicker than '2 mm.; it must be perfectly clean, and before use be heated to a red heat in the Bunsen flame. Adjust the height of the burner so that the hottest part of the flame, which is just above the top of the inner cone, lies about the level of the lower portion of the slit. Place a small quantity of barium chloride on the platinum wire, wetting the latter first with a little dilute hydrochloric acid. Support the wire in the stand provided and insert it into the hottest part of the flame. After a few seconds the upper part of the flame will be green coloured, and on looking through the spectroscope you should see a spectrum of bands, with a line in the green which is sharp and bright. If this line XXXIX MAPPING SPECTRA 186 does not appear, the adjustment of your spectroscope is probably faulty or your slit is too wide. If the spectrum is very faint, either the flame is not placed properly in front of the slit, or the latter is too narrow. The best width of slit is only obtained by trial. For the observation of the spectra of the alkali metals with small spectroscopes, the slit ought not to be so narrow that the sodium line appears double, but it ought to appear a sharp line and not a broad band. Represent on the paper scales provided, by meaus of shading with a sharp black pencil, the appeai*ance of the bands and lines seen in the spectroscope. After you have made an independent drawing, consult some atlas of spectra and compare yours with the drawing given there. If you tind that you have not seen all the bands shown there, repeat your experiment. Some of the fainter bands may however escape your notice, as they require a completely darkened room. Make similar drawings of the spectra of lithium, thallium, strontium, and calcium chlorides. Notice that the spectrum of the last-mentioned salt gi-adually changes. The spectrum first seen is due to the chloride of calcium, and is gradually replaced by that of the oxide. Make separate drawings of the first and last stages through w^hich the spectrum passes. During the observations, check from time to time the position of the sodium line. If you find that a slight change has taken place, it is not necessary to alter the position of the scale-tube, apply instead a correction to your readings. Thus, if the sodium line reads 8*1 instead of 8, you may correct all other readings by subtracting '1. As a last example take a bead of potassium chloride. You should have no difficulty in seeing a line in the red, which is really a double line. But there is also a line at the extreme end of the visible violet, which requires special precaution to see. Not to have your eye disturbed by extraneous light, move the telescope until the red and green are out of the field of view, so that when looking at the continuous spectrum of a gas flame you only see the violet ; also remove the burner which illumi- nates the scale. Carefully introduce the potassium salt into the hottest part of the flame while you are looking through the 186 LIGHT XXXIX telescope ; -svith a little practice this is easily done. If you fail to see the potassium line, slightly open the slit. As soon as you can see it, illuminate the scale a little and take a reading. Finally, move the telescope back, and without altering the width of the slit, take a reading of the sodium line. The measurement of the potassium lines should be made several times in order to secure greater accuracy. It will be noticed that while thallium, sodium, potassium, and lithium give line spectra, the spectra of the alkaline earths consist of bands. These latter are due to the metallic oxides or chlorides, while in the case of the alkali metals the salts are decomposed and the spectra of the metals themselves appear. Instead of making shaded drawings of the appearance of spectra, they may be represented diagrammatically, as in Fig. 80, where the intensities are represented by the lengths of the lines. Thus 5 is a sharp line, the intensity of which is estimated to be about J that of J., i) a band having a sharp edge at 11*30 but fading away gradually to the other "pig, 80. side, the limit of visibility being about 11-7, G, on the other hand, a band which is brightest in the middle and ends abruptly at 13-6 and 13*84. Iiiiilmilii| i l|inlii'iliiiiliii!liri SECTION XL. REDUCTION OF SPECTROSCOPIC MEASUREMENTS TO AN ABSOLUTE SCALE. When the spectrum of a given substance is mapped, the relative positions of the lines will be found to differ according to the instrument used. Even if the sodium line is at the same scale division in each, the readings of the other lines will differ in the different instruments, since they depend on the dispersion of the prism and on the distance between the divi- sions of the scale. In order that measurements taken with different instruments may be comparable, it is necessary to have some method of representation which is independent of the instrument employed. It is usual to take the wave-length as that which character- ises a given light vibration, and the spectrum mapped according to the wave-length scale is called a ** normal " spectrum. In- stead of the wave-length scale, we might adopt one which takes the " time of vibration " as the characteristic property of a set of waves, and such a scale would present many advantages. The time of vibration t is connected with the wave-length X by the relation X = t;f, where v is the velocity of light. If a set of waves in air, from a source sending out homogeneous light, enters a different medium, as for instance a glass prism, the time of vibration will be the same in the glass as in air, but the velocity of propagation of the waves, and hence the wave-length, will be different. When a particular kind of homogeneous light is defined by its wave-length, we must therefore state the medium in which that length is measured. 188 LIGHT . XL The most scientific proceeding would be to take the wave- length in vacuo as the standard, but as all direct measurements of wave-lengths are taken in air, and the reduction to vacuo involves an accurate knowledge of the refractive index of air, it is usual to take the wave-lengths in air as standards. To be strictly accurate we must define the pressure and the tempera- ture of the air, since its refractive index depends on its density. Rowland's table of standard wave-lengths is now generally adopted, and holds for a pressure of 76 cms. and a temperature of 20° C. In certain investigations it is necessary to know the wave-lengths in vacuo, and Tables have been constructed ( Watts's Index of Spectra, Appendix E) by means of which the conversion to vacuo may be easily effected. In order to avoid having the decimal point at an incon- venient place, wave-lengths are usually given not in centimetres but in so-called tenth-metres (written X"' metre) the lO^^'th part of a metre or the lOHh part of a centimetre. According to this scale the wave-lengths of the two sodium lines in air are 589018 and 5896"15, and the correction to vacuo is + 1*60. Spectroscopes containing one prism of the usual size only barely separate the sodium lines, and measurements with such instruments can only be accurate to two or three X*^'^ metres. All figures beyond the decimal point may therefore be left out of account in what follows. It is sometimes advantageous to consider the " wave- numbers," or namher of waves per centimetre rather than the wave-lengths. If there are v waves in one centimetre z^A,= l, and hence the wave-number is simply the reciprocal of the wave-length. For the least refrangible sodium line the wave- number is therefore Different methods may be used to reduce measurements taken on an arbitrary scale to wave-lengths, and it is necessary to gain practical experience of the most important of these. In each method the positions of two or more lines the wave- lengths of which are known, are observed on the arbitrary scale of the spectroscope, and the unknown wave-lengths of other wave XL REDUCTION OF SPECTROSCOPIC MEASUREMENTS 189 lines obtained from them by interpolation. It is in the process of interpolation that the methods differ. Method I. Graphical inte-rpolation by means of a wave- length curve. When the same spectroscope is always used by the same observer and its adjustments remain unaltered, it is generally simplest to draw once for all a curve connecting the scale read- ings with the wave-lengths. To do this, the positions of a certain number of so-called "standard lines" are observed on the spectroscope scale, and a curve having the scale divisions as abscissae and the wave-lengths as ordinates is drawn. The shape of the curve will resemble that shown in Fig. 81. It is important to choose the scale of the curve suitably. If the paper is divided into millimetres, each scale division may be made to cor- ^^^ respond to one millimetre, as it will enable subdivisions to be estimated to the same degree of accuracy on the curve as in observing a line. As regards the scale of ordinates, it must be remembered that the visible spectrum ranges from 7,900 to 3,900 X'** metres, and that it will not be possible to determine with an ordinary one-prism spectroscope the last figure accurately. Hence if one millimetre is made to correspond to 20 X'** metres, the scale will be suffici- ently large, and the total length of paper necessary for the curve will be 20 centimetres. Once the curve is drawn the wave-length of any unknown line may be obtained immediately by inspection. A list of convenient standard lines is given at the end of this section. Exercise I. Draw a curve connecting the wave- lengths with the scale readings of a spectroscope. From the observations you have made of the lines whose wave-lengths are given in the Table at the end of the section, draw the curve a« described, and prepare a Table giving the wave-length for each tenth scale division of the instrument you have used, thus : Scale reading Fig. 81. 190 LIGHT XL Spectroscope A. Sodium lines placed at 8. Scale Division. Wave-Lengths in Xth metres. 4 7862 5 7239 6 6669 7 6281 8 5893 9 5622 10 5351 11 5180 &c. &c. The Table should extend over the whole of the visible spectrum. Method II. Graphical Interpolation by means of a curve of wave-numbers. The curve given in Fig. 81 has a some- what large curvature, hence the number of standard lines required to draw it accu- rately is comparatively large. To obtain a curve w^hich is more nearly straight, and can therefore be drawn with greater accu- racy, we make use of the wave-numbers v as Fig. 82. ordinates instead of wave-lengths X (Fig. 82). To find the wave-length of an unknown line its scale reading is taken and the corresponding value of v found by the curve. The required wave-length will then be the reciprocal of the value found. The wave-numbers per centimetre of the red and the blue potassium lines are respectively 13016 and 24717. If therefore, each millimetre of the squared paper corresponds to 50, the vertical distance required will be 24 cms. Exercise II. Draw a curve connecting the inverse wave-lengths or wave-numbers with the scale readings. Prepare a second Table from this curve, thus : — XL REDUCTION OF SPECTROSCOPIC MEASUREMENTS Spectroscope A. Sodium lines at 8. 191 Scale Division. Wave-Length. cms. Wave-Numbers. 4 5 6 7 8 7-860 X lO-** 7-238 6-712 6-277 5-893 (fee. 12723 13816 14895 15931 16969 <fec. Method III. Graphical interpolation by means of inverse squares of wave-lengths, or sqvared wave-numbers. This method will be found to give a curve very nearly straight, but for the present purpose the increased amount of calculation involved in the method renders its use incon- venient. Method IV. AHthmetical interpolation of wave-lengths. It is sometimes possible to find two known lines near to, and one on each side of, the line to be measured. Calling the wave-lengths of these two lines X, and X^ and their scale readings 71, and n^, it is seen that a distance of n^ — n^ on the scale corre- sponds to a change of wave-lengths X, — X^. Hence if n is the reading of the unknown line and X its wave-length, a difference n — ni on the scale must correspond to a difference X, — X in wave-length, and if change in wave-length is proportional to change in reading Xi — X n — ni Hence A«j "^ A^ n^ "^ Tlj an equation which we may use to calculate X when the three lines are near together, but not in other cases. Method V. Arithmetical interpolation by means of wave- numbers. The use of this method will be understood from what pre- cedes. 192 LIGHT XL Method VI. Arithmetical interpolation by means of inverse squares of wave-lengths or squared wave-numbers. It has already been stated that equal distances on the scale correspond very nearly to equal differences in the inverse squares of the wave-lengths. Hence calling 3/1, y^ and y the inverse squares of the wave-lengths of the two reference lines and of the unknown line respectively, and replacing the values of \ by the corresponding values of y in the result given in Method IV. we have : — 2' = 2/. + ;^^/2/3-2/.). From y, the wave-length is found by calculating Ijs/y. When the reference lines are not very close together this is the method which should always be adopted when arithmetical interpolation is employed. Exercise III. Map a spectrum on the normal scale. Using the curve obtained in Exercise XL, make as accurate a drawing as you can of the calcium spectrum on a scale of wave- lengths, representing the characteristic features of the spectrum by one of the two methods of mapping spectra described on pages 185 and 186. Exercise IV. Using the measurements previously made, calculate, by Methods IV. and VI. above, the wave-length of the sodium line, having given the wave-lengths of the lithium and thallium lines. Enter observations and results as follows : — Line. Scale Division. Wave-lengths in Xtt metres. Given. Calculated Method IV. Calculated Method VI. Lithium 6-15 6708 Thallium . . . 10-55 5351 Sodium 8-0 5932 5898 Correct Wave-length = 5893. XL REDUCTION OF SPECTROSCOPIC MEASUREMENTS 193 Notice that Method IV. gives only an approximate value, but that Method VI. gives a fairly accurate result. This shews that the reference lines are too distant fi*om the unknown line for the first method of direct interpolation to be applicable, and for very accurate work even the method of inverse squares should not be used for lines which are as wide apart as these. TABLE. Wave-length Wave-number Inverse Square of Reference Lines. Xth Metres l>er Centimetre Wave-length in air. in air. in air. K red. Centre of 7682-5 13016-6 1694-3 X 10» double line Li red 6708-2 14907-1 2222-2 H. red. (C) 6563-05 15236-8 2321-6 Na yellow. Cen- 5893-17 16968-8 2879-4 tre of double line (D) Tl green. 5350-65 18689-3 3492-8 Up green (F) 4861-50 20569-8 4231-2 Sr blue 4607-51 21703-6 4710-5 Ca blue 4226-90 23658-0 5597-0 K violet. Centre 4045-82 24716-9 6109-3 of double line The wave-lengths and wave-numbers are here given to six significant figures, as the more complete values may be useful for reference, but for the purposes of the above exercises it will be sufficient to use four figures. The wave-lengths are taken as far as possible from Rowland's maps; in the case of Potassium, Lithium, and Thallium the numbers are those given by Kayser and Runge. 8. P. 13 SECTION XLI. THE SPECTROMETER AND ITS ADJUSTMENTS. Apparatus required : Spectrometei\ with Gauss eye-piece, plane parallel glass on stand. An instrument in which the deviation produced by the passage of a beam of parallel light through a prism or other apparatus can be measured is called a spectrometer. It consists of a collimator S, a telescope T, and a horizontal divided circle G (fig. 83). The telescope moves about a vertical axis passing through the centre of the circle. Attached to the telescope is an arm carrying two verniers V, V opposite Fig. 83. each other, which move along the circular scale, so that the angle through which the telescope is turned may be accurately measured*. Above the divided scale is a table B, on which prisms or gratings may be placed. In some instruments * In fig. 83 the circle and verniers are enclosed in a case. XLI THE SPECTROMETER AND ITS ADJUSTMENTS 195 this table is fixed, but ought to be capable of being turned about an axis coincident with the axis of rotation of the telescope. The way in which an angular displacement of the table is measured diffei-s in different instruments. If the divided circle is fixed to the collimator, the table should carry a second set of verniers. But in many instruments the table and divided circle move together. If that Ls the case, care must be taken that between any two readings of the verniers either the table only, or the telescope only is moved. If both are displaced, the vernier readings will only shew the difference between the angular displacements of table and telescope, not the actual displacement of either. It is difficult to secure that the axis of rotation passes accurately through the centre of the divided circle. The error thereby introduced, called " error of eccentricity," is eliminated by having two verniei-s opposite each other as described. The angle measured by means of one vernier exceeds the correct value by as much as the angle measured on the other falls short of it ; so that the arithmetic mean of the two results gives the correct angle. For a proof of this see Note, p. 202. Examine the scale and vernier, and determine the value of the smallest subdivision of the principal and vernier scales. If e.g. the circular scale is subdivided into 20 minutes of arc, the vernier will probably be divided into 20 parts, of which every fifth will be numbered, and each of the 20 may be sub- divided into 2 or 3 parts, which will allow the measurements to be made to oO or 20 seconds of arc. In entering an obser- vation in your Note-book, write down separately the readings of the two scales. Thus if the angle to be read off were At*!" 43' 20", the observation would be entered as follows: — Principal Scale 47° 40' 0" Vernier 3' W 47° 43' 20" A w(M^l('u model ot the vernier is placed in the laboratory, and the student should practise reading it until he is quite familiar with it. Before proceeding with the exercises students should study 13—2 196 LIGHT XLI the construction of the instrument they are using, and refer to a more detailed description, which will be supplied with the instrument. Special care should be taken to be familiar with the object of the various screw heads, which serve either to clamp some part of the instrument, or to give that part a slow motion. If the telescope, for instance, is to be pointed in any direction, it is first moved by hand as nearly as possible into its right position. It is then " clamped " by the proper screw, and finally brought to the proper position by means of a " fine adjustment screw," which can, within certain limits, alter its direction. For the purpose of clamping, it is not necessary to use any force. If tlie screw is screwed up gently it will he sufficient. If force is used the instrument will be damaged. Other moveable parts of the instrument will also in general be provided with a clamping arrangement and fine adjustment. Before moving any part of the instrument care must be taken that the clamping screw of that part is released. All parts should move easily, and no force shoidd ever be used. In order to fix the direction in which the telescope points, a mark is placed in the focal plane of the object lens. This mark consists generally of a cross formed by two cocoon fibres, spider's threads, or very thin platinum wires. This arrangement, called the " cross wires," can be turned in its own plane, so that one of the wires may be placed vertically, or they may both be placed at an equal inclination to the vertical. The latter position is most convenient when the telescope is to be pointed towards the image of a slit, for it is found easier to. Fig. 85. place the centre of the cross on the image when the cross has the position shewn in Fig. 84, where AB represents the image XLI THE SPECTROMETER AND ITS ADJUSTMENTS 197 of the slit, than when one of the wires is vertical, as in Fig. 85. For when the slit is narrow, as it should be, the vertical wire will be difficult to see against the dark background. When the field of view is dark, it may be necessary to throw some light into the telescope in order to see the cross wires distinctly. This is best done by means of a sheet of white paper held in the hand and placed obliquely near the object- glass of the telescope ; the light scattered from the white paper will illuminate the field of view sufficiently. To focus the eye-piece on the cross wires, there must be some means of altering the positions of one of the two. If the cross wires are fixed in the draw tube, the eye-piece may be moved slightly outwards or inwards, if the eye-piece is fixed the wires are mounted on a frame which admits of a small motion. Turning the telescope towards a bright surface, such as an illuminated sheet of paper, move the eye-piece or the cross wires until you see them distinctly. The eye will be least fatigued, if the cross wires are so placed that their image is as far removed i.e. the eye-piece drawn out as much as possible, consistently with distinct vision. To focus the telescope turn it towards some distant object, and alter the distance between object lens and eye-piece till a distinct image of the object is produced at the cross wires. Move the head sideways to see that there is no parallax. If a parallel beam entei-s the telescope in such a direction that it converges to the central point of the cross wires, a line drawn through that central point parallel to the original direction of the beam, is the " line of sight " of the telescope. If the light comes from a star, the line of sight is the line drawn from the centre of the cross towards the star. This line of sight should be perpendicular to the axis of rotation of the telescope (see p. 199). In order to adjust the collimator to give a parallel beam of light, tuni the telescope so as to look straight into the collimator tulie. Place a luminous burner 10 cms. behind the slit of thr latter and focus the telescope until the edges of the slit look perfectly sharp. Place the cross wires on it, move the eye to the right or left and see that thfere is no parallax. 198 LIGHT XLI The level of the collimator must now be fixed, so that a point near the centre of the slit shall have its image covered by the cross wire. For this purpose a fine wire or thread should be stretched across the centre of the slit. If the level of the collimator admits of alteration, it should be adjusted until the image of this thread coincides with the centre of the cross. If the collimator is fixed, the position of the thread on the slit must be altered until the condition is nearly satisfied, great accuracy of this adjustment not being important. Another method of focussing telescope and collimator is sometimes found more convenient, as it does not involve removing the v prism or other apparatus from the ^', ^ j^ table. This method is based on the i ,.^--<^i^^^^^^~"^<> fact, that if a beam of light diverging S |-^*''^^^^^ from a point S falls on a prism, the ' distance of the virtual image S' ^^^- ^^• from the prism is greater the greater the angle of incidence of the original beam. If the prism is placed in the position of minimum deviation (Fig. 86), the virtual image will be at the same distance as the object, but as the prism is turned in the direction of the hands of a watch, the virtual image will move further away. If the distance of S from the prism is very great, so that a parallel beam of light falls on the prism, the rays will remain parallel after refraction, whatever the angle of incidence. If the original beam is parallel therefore, and only in that case, will the position of the prism have no effect on the distance of the virtual image from the observer. This furnishes a delicate test of the parallelism of a beam of light. If the light is not parallel the adjustment may be made as follows : — Place a prism on the spectrometer table approximately in the position of minimum deviation. Illuminate the slit with sodium light and turn the telescope so as to see the sodium line. Fix the position of the telescope so that when the prism is turned round, the image moves across the field of view and comes to minimum deviation just at its edge or a little beyond. It is clear that there will now be two positions of the XU THE SPECTROMETER AND ITS ADJUSTMENTS 199 prism for which the sodium line will be in the centre of the field of view. In fihe first position, which we may call the " slanting position," the incidence of the beam on the prism is greater than at minimum deviation, while in the second which we may call the " normal position " the incidence is smaller. Rotate the prism to the slanting position, and focus the telescope carefully until you see the edges of the slit quite sharply ; it is better for this purpose not to have too narrow a slit. Now turn the prism into the normal position. If the focus is still good, the collimator supplies a parallel beam. But if the image is out of focus, this shews that the collimator is not in adjustment, and as we know that the change of position of the prism must have brought the image nearer, we may remove it to its old position by adjusting the collimator. When this is done the prism is turned bacj^ to the slanting position. In doing so we remove the image still further, and must focus again, but this time with the telescope. Repeat the process several times, taking care always to focus the telescope tuhen the prism is in the slanting position, and the collimator when the prism is in the normal position. After three or four changes it will generally be found that the image remains in focus during the displacement of the prism, and the adjustment is then complete. If the telescope is now directed straight towards the collimator, the image of the slit should be in focus at the cross wires, and there should be no parallax on moving the eye to the right or left. When the prism is in use, if the faces of the prism are not quite plane, it may happen that the focus is slightly different according as light passes through the prism, or is reflected from one of its faces. Unless the image is rendered markedly indistinct it is best, however, not to alter the focus. The prisms generally supplied with spectrometers of ordinary size are sufficiently plane not to require any readjustment of focus. We have seen that the line of sight of the telescope should be at right angles to the axis of rotation, i.e., parallel to the plane of the divided circle. This adjustment is more difficult to carry out. Fortunately 200 LIGHT XLI the errors introduced, when the condition is not accurately fulfilled are not very great, and unless greaBaccuracy is required it need not be carried out. The m^thod^ instructive however, and should be understood. ^ In Fig. 87 let AB he the axis of rotation, LM the line of sight of the telescope. Also let UK be a piece of plane parallel glass; placed so that a ray of light passing along ML is reflected back along the same line LM, i.e. with HK at right angles to Fig. 87. LM. If now the arm OGD is rotated through two right angles round AB, until the line of sight comes into the position L'M\ an inspection of the figure will shew that the surface HK will no longer be at right angles to L'M\ unless the line LM was originally at right angles to AB. It remains to be shewn how the line of sight of the telescope may be adjusted to satisfy this condition. The telescope is provided with an eye-piece having an aperture at the side, through which light may fall on an inclined piece of glass inserted between the two lenses of the eye-piece. The light from a luminous burner is reflected from the glass plate, and illuminates the cross wires (Fig. 88). ^^ Fig. 88. The light diverging from these wires is converted by the object-glass of the telescope into a parallel beam of light, XLI THE SPECTROMETER AND ITS ADJUSTMENTS 201 which is reflected from the plane surface HK along its own path, if that surface is at right angles to the line of sight of the telescope. An image of the cross wire will be formed, which will coincide with the wires themselves if all the adjustments are perfect. A slight displacement of HK will displace the image of the cross wires, and unless HK is approximately in the right position, the image may not appear in the field of view at all. In making the adjustment proceed as follows. Place the glass plate on the table of the spectro- meter so that the plane of the plate is perpendicular to the line joining two levelling screws of the table, and loosen the clamping screws of table and telescope. Turn the table until the line of sight of the telescope is, as far as you can judge by the eye, at right angles to the glass plate. Place your eye near the eye-piece of the telescope, and look at the image of the telescope in the glass plate. This image should be in line with the telescope itself as in Fig. 89. Fig. 89. Fig. 90. If the image points downwards, as in Fig. 90, the plate should be tilted backwai-ds by means of the levelling screws. A little practice will enable you to set the plate nearly right in this way by eye. When this is done, clamp the table, place a light at the side of the eye-piece, and look through the telescope. A small angular movement of the telescope to the right or left; should now bring the image of the cross wires into sight. Now level the glass plate till on moving the telescope the image of the cross wires coincides exactly with the wires them.selves. The adjustment having been made on one side, turn the telescope* through two right angles, and see whether the image of the cross wires formed by reflection from the other side of the plate coincides with the wires. If this is the case, the line 202 LIGHT XLI of sight is at right angles to the axis of rotation as required. If not, bring the two images into coincidence by first tilting the telescope until their distance apart has been halved, and then tilting the glass plate until the images are in coincidence. Turn the telescope again through two right angles, and if necessary repeat the adjustment till the images coincide with the cross wires in both positions of the telescope. If the two images of the cross are not both sharply defined, the telescope has not been properly adjusted for parallel rays, or the glass plate is not perfectly plane. In Note. In Fig. 91 let G be the centre of the divided circle, any position of the telescope let one of the verniers be at A and the other at A\ and let be the axis of rotation of the telescope. Let the telescope be rotated till the verniers are at BB\ Then BOA will be the angle through which the telescope has been rotated, while BCA and B'CA' will be the apparent angular displacements of the two verniers as read off on the divided circle. But BCA = 20B'A = BOA + OF A - OAF and B'CA' = 20 AF = BOA + OAF - OF A Hence by addition BCA -{-B'CA' = 2B0A which shews that the ^ngle thi'ough which the telescope is rotated, is correctly obtained by taking the arithmetic mean of the angles read off on the vernier scale. SECTION XLII. DETERMINATION OF THE REFRACTIVE INDEX OF A SOIJD BY THE SPECTROMETER. Apparatus required : Spectrometer in adjtcstmerUy glass prism, small mirror and reading lens, sodium bead and flame. The method used in determining the refiuctive index of a solid is identical with that explained in the elementary exercise * on the same subject, with which the student is supposed to be familiar. The first step is to determine the refracting angle of the prism. For this purpose the prism must be placed on the table of the spectrometer so that its two refracting fiices, and therefore their intersection, are perpendicular to the plane of the graduated circle. The table of the spectrometer ought to consist of a platform which can be levelled independently of the circular scale. If this is not the case, the prism should be placed on a separate support provided with levelling 8cre^vs. In Fig. 92 PQR are the three *'»«• ^^' levelling screws, and the prism is placed in the centre of the table in such a way that one of the faces, say AC, is at right angles to the line joining two screws, say R and Q. This may be done with sufficient accui-acy by eye. To adjust the faces of the prism, place the collimator and telescope at a small angle to each other, illuminate the slit and turn the table of the spectrometer until an image of the slit, formed by reflection at AC, appears in the centre of the field. • IfUermediaU Practical Phy$icM, Sec. xxviii. 204 LIGHT XLII The slit should be sufficiently narrow to allow you to place the cross accurately over its image ; but it is not necessary to have it as nan-ow as you would use it in spectroscopic work, unless you desire to resolve the double sodium line and measure the refractive index for each line separately. The spectrometer table must now be levelled so that the image of a thread stretched across the centre of the slit coincides with the centre of the cross wires of the telescope. The table is then turned till the reflection of the slit is obtained from the face AB, and the image of the thread made to coincide with the centre of the cross wires by levelling the prism by means of the screw P, which is the only one that does not alter the inclination of the face AG, but only turns that face in its own plane. After the face AB has been levelled in this way, return to the face AG; if you have not originally succeeded in placing AG at right angles to QR, you may have slightly altered the inclination of the face. If so, set it right again by the screw Q which least disturbs the face AB. Going backwards and forwards once or twice will always enable you to secure that the two faces are both properly adjusted. Proceed next to measure the angle of the prism. Keep the telescope fixed, and turn the prism so that the intersection of the cross wires lies exactly on the image of the slit formed by reflection at one face of the prism. Read both verniers. Next turn the prism till the image of the slit formed by the second face is on the cross wire, and again read both verniers. If the table has been turned through an angle 0° between the two observations, the angle of the prism is 180° — 6°. If in moving from one position to the other, a vernier passes over the zero of the scale, 360° must be added to the smaller reading, and the higher reading subtracted from it. Alter the position of the telescope by a few degrees, and repeat your observations, obtaining a second value for the angle of the prism, which should agree closely with the first. This method implies that the table of the spectrometer is moveable. If it is not so, the method described in the Note p. 207 must be adopted. Vernier A. Vernier B. 350° 6' 0" 170° 2' 20" 110° 13' 0" 290° ir 0" 120° r 0" 120° 8' 40" ... *- 120° r 50" 59° 52' 10" XLII REFRACTIVE INDEX OF A SOLID 205 Enter your observations as follows : Jan. 24, 1894. Spectrometer J., Prism A. Determination of Angle of PHsm. First position of prism Second „ Difference ... Mean .*. Angle of prism ... Similarly for the second measurement. Having determined the angle of the prism proceed to find the minimum deviation for light of the kind for which the refractive index of the prism is required, e.g., sodium light. Place a Bunsen burner with a sodium bead in it behind the slit, and without altering the adjustment of the prism turn the spectrometer table so that the light coming from the collimator is refracted through the prism. Find the refracted image of the slit in the first place by eye, and follow the image while the prism is slowly turned round till the position of minimum deviation is found roughly. When this is done bring the telescope into this position, and find the image of the slit in the telescope. Watch this image while the table of the spec- trometer is slowly turned. If the direction of rotation is ))roperly chosen, you will find the image moves slowly in the direction of deviation, comes to a standstill, and moves back again. Leave the prism in the position for which the deviation is least, place the cross wires of the telescope approximately on the image, clamp* the telescope and adjust the cross wires more accurately by means of the slow motion. Now tuni the prism again backwards and forwards, making sure that the centre of the image of the sht just comes up to the centre of the cross wires, but does not pass beyond it. Adjust the telescope if nece.ssary until you are quite satisfied that this is the msv. Rriul both Verniers. ' Sro ifiuuiks lis t(j clamiung, Ac, p. i'.M). 206 LIGHT XLII Now clamp the table of the spectrometei'. Remove the prism, turn the telescope so as to point directly towards the colli- mator, and adjust it until the image of the slit is bisected by the cross wires. Read the verniers to determine the position of the telescope when there is no deviation of the ray. The difference in the readings in this position and in the position of minimum deviation gives the angle of minimum deviation. It is advisable to obtain two independent readings of the position of minimum deviation so as to secure accurate results. When the spectrometer table is fixed, it is not material whether the direct reading is taken before that of minimum deviation or vice versa. But when the graduated circle moves with the spectrometer table the above order mitst be adhered to. For it has already been pointed out that correct values will in that case only be obtained if either the telescope only, or 'the table only, is turned round between two readings. As the adjustment to minimum deviation necessarily involves the turning round of the table, this should be done first and the table be clamped before the telescope is moved round to take the direct reading. The observations are entered as follows : — Detertnination of Minimum, Deviation. Vernier A. Vernier B. Prism at minimum deviation 143° 40' 00'' 323' 40' 40" Direct reading 94° 13' 00" 274° 14' 20" 49° 27' 00" 49° 26' 20" Angle of minimum deviation 49° 26' 40" Similarly for the second experiment. Angle of minimum deviation 49° 27' 40" MeanD 49° 27' 10" Angle of prism a 59° 52' 10" D + a = 109° 19' 20" (D + oi)/2 = 54° 39' 40" a/2 = 29° 56' 05" . n + a sm — ^^ ^.aoA^ a sm 2 XLII REFRACTIVE INDEX OF A SOLID 207 The observations should be suflSciently accurate to give the third decimal place with certainty, and the fourth decimal place with fair accuracy. The numerical calculation should be carried out with a table of seven figure logarithms. Calculate the error produced in n on the supposition that an error of half a minute has been made, (a) in the measure- ment of the angle, (6) in the determination of the minimum deviation. Note. If the table of the spectrometer is not moveable, the angle of the prism may be measured as follows: — Place the prism with its refracting edge towards the collimator in such a position, that about half the beam of light falls on one and half on the other face of the prism. A reflected image may now be obtained from each face, and if the telescope is first placed so as to point to one image, and then turned until it points to the other, the angle through „. ' ' which the telescope has been turned is twice the angle of the prism. In Fig. 93, let / be the virtual image of the slit formed by the collimator, ABC the prism, /, and Li the two images formed by reflection at the faces AB and AC oi the prism. By the laws of reflection we have the angles lAI, = 2IAN, lAI, = 2IAN, .'.I,AI,--2N,AN, = 2a. The images /, and /j therefore subtend an angle 2a at the refracting edge A, and if the telescope is pointed first towards /, and then towards I^, the angle through which it turns will be 2a, if the axis of rotation of the telescope is at A, or if / is at an infinite distance. Hence, in order that this method should give correct results, the collimator ought to be well adjusted for parallel light, and to correct any error due to a faulty adjustment, the refracting edge should be placed nearly over 208 LIGHT XLII the axis of rotation of the telescope. The errors introduced may otherwise be quite appreciable. If, for instance, the distance between the refracting edge and the axis of rotation is one centimetre and the image of the slit at a distance of 50 metres, which is quite possible if the collimator is of ordinary size, the error may amount to over a minute of arc. Quite apart from this source of error, which might be avoided, the method is not a good one, for each image is formed by an unsymmetrical beam of light, filling at the utmost only half the telescope lens, and serious errors may creep in due to the aberrations of the lenses. SECTION XLIIL DETERMINATION OF THE INDEX OF REFRACTION OF A LIQUID. SPECIFIC REFRACTIVE POWERS. Apparatus required : Spectrometer, hollow glass prisin, alcohol, theinnometer, Bunsen flame, and sodium head. When the index of refraction of a liquid is to be determined, the liquid is placed in a hollow prism the vertical sides of which are two plates of glass. If these glass plates are accu- rately plane parallel they produce no deviation of the rays of light passing through them, and the deviation observed is due to the liquid only. Hence the angle of the prism of liquid may be measured, and the refractive index determined by the devia- tion, just as in the case of a solid glass prism. If the plates of glass are slightly prismatic their effects on the apparent angle and deviation must be eliminated. To enable this to be done they are not cemented to the rest of the prism, but are made moveable, and are kept in position by rubber bands or by clips. After observations of the angle of the prism and of the deviation have been made each plate is rotated about its normal through 180^ and the observations repeated. The means of the two sets of observations of the angle and of the deviation are the required angle and deviation of the liquid prism. Clean the prism provided, and fill it with absolute alcohol at the temperature of the room. As the prism may leak a little, it is not placed directly on the table of the spectrometer, but in a shallow metal dish which stands on three short legs on the table, and itself supports the prism at three points. The adjustment of the prism and the determination of the angle and deviation are made as in the previous section. In 8. P. U 210 LIGHT XLIII determining the angle the stronger of the two reflected images of the slit seen in the telescope should be used in each case. It is formed by reflection at the air-glass surface, while the weaker is formed by reflection ai the glass-liquid surface. The glass plates forming the sides of the prism are then rotated as described above and the observations repeated. Each plate is provided with a mark, the position of which should be recorded, along with the observations. The mean deviation D and the mean angle a are then found and the refractive index calculated from them in the usual way. Enter your observations and results as follows : — 15 October, 1895. Prism 1 1. Absolute Alcohol. Determination of deviation produced by prism when the marks on the glass plates are near angle of prism : — With prism Without „ Apparent deviation Determination of angle of prism when the marks on the plates are near angle of prism : — First position ... 246° 29' 30" 66° 29' 30" Second „ ... 126 29 306 29 30 Difference ... 120 30 120 Mean 120° 0' 15". .'. Apparent angle of prism = 59° 59' 45". Give similarly the readings of the verniers when the plates have been rotated and the marks are near the base of the prism, and collect the results as follows : — Mean minimum deviation = D Mean angle of prism = a . • . Refractive index Temperature of alcohol Refractive index calculated from re- Vernier A. Vernier B. 229° 5' 0" 49° 4' 30' 23 9 203 7 30 25° 56' 0" 25° 57' 0" ... 25° 56 ;' 30" 25° 36' 30" 59 51 42 1-3635 ... 18° C. f ,. , 1-365 fractive powers XLIII REFRACTIVE POWERS 211 If greater accuracy is requii-ed several measurements should be made and the mean of the results taken. The observed refractive index may be compared with the index calculated by a method founded on the following facts : — If fi is the refractive index and d the density of a substance, the value of the expression , is called by Gladstone the specific refractive power of the substance. It is not sensibly altered by variations of temperature or pressure, but changes when the substance passes from the liquid to the gaseous state. Lorenz and Lorentz have from theory deduced another expression, namely, -, — ^--t. which remains nearly constant, even when /I* 4- 2 a the substance changes its state. For the purpose of this exercise it will be sufficient to use the former expression, as it more easily lends itself to numerical calculation. The refrac- tive power varies with the wave-length of the light used, and we shall take ^ to refer to soilium light. It is found by experiment, that the refractive power of a mixture of substances is equal to the mean refractive power of the constituents. Thus, if p^, p^, p^, &c. grams, of a number of liquids are mixed together, we have where P = pi-\-p.,-\-p:i+ ... , and (N — I )/D is the refractive power of the mixture. It has also been found, that for a compound, the molecule M of which contains n^ atoms of an element the atomic weight of which is m,, /i, atoms of an element the atomic weight of which is jti-i, and so on, the weights of the different elements in the combination being therefore Tiiniit w,?/*,, etc., we have iX ^ \ uu — 1 ii. — 1 If we call ^ J - the atomic refractive power of the first element, and designate it by r^, and so on for the other elements, 14—2 212 LIGHT XLIII f,-l a the molecular refmctive power R of the compound, we have then the equation R = 7ii?'i + iur._, + &c., or in words : — the refractive power of a compound is the sum of the products formed by multiplying the atomic refractive power of each element by the number of atoms of that element con- tained in the molecule of the compound. We cannot dii*ectly determine the atomic refractive power of an element entering into a compound, but we may find it in- directly. If, for instance, we take three compounds containing carbon, hydrogen, and oxygen in different proportions, we obtain three equations to determine ?\, r^, and r^ the atomic refractions of carbon, hydrogen, and oxygen respectively, and if we find that the refractive indices of other compounds may be determined with the help of the values so found, we shall have justified the above statement. It appears, however, that the way in which an atom is combined affects its refractive power, and to obtain consistent results it is necessary to assume, for instance, that oxygen in the carbonyl group has a different refractive power from the hydroxyl oxygen. Calculate the refractive index of ethyl alcohol, making use of the following numbers : — Atomic refraction of ^ ... ... ... 1*66 „ „ of in alcohols ofC Density of ethyl alcohol, at 20° = Coefficient of cubical expansion = Calculated index of alcohol at 18° C. = Measured „ „ = 2-68 4-36 •788 •0010 1-365 l-3(>3 SECTION XLIV. PHOTOMETRY. Apparatus required : Photometer, scale, candles, and gas- lamp. The photometer is an instrument for comparing the illu- minating powers of different sources of light. One of the most accurate forms of the instrument is that in which the two sources are placed on opposite sides of a screen capable of partly reflecting and partly transmitting the light from each source. When this screen is placed in such a position that the straight lines drawn from it to the two sources make equal angles with the screen, and the illumination of the two sides of the screen appears the same, the amounts of light to be compared are to each other as the squares of the distances of the sources from the screen. Having fixed on the light sent out by some standai*d source in a given direction as the unit, we can by a comparison of any source with the standard determine the illuminating power of that source. The standard source is a .sperm candle flame consuming 120 grains of sperm per hour, and the standard direction any line in a horizontal plane. If the consumption of sperm differs from the stjindard mte by less than 10 grains per hour, the illuminating power may be taken as proportional to the con- sumption. Arrange the two sperm candles, the photometer, and the gas flame provided, in a straight line in the order named, the 214 LIGHT XLIV two candles being mounted about 5 cms. apart in a direction perpendicular to the straight line, on a stand capable of moving along a graduated scale parallel to the line, Fig. 94. Place the Fig. 94. gas flame at some convenient fixed distance from the photo- meter screen, say 1 metre, with a flat surface towards the screen, light the candles and after a few minutes move them backwards and forwards along the scale, till the illumination of the two sides of the photometer screen is equal. If the simple Bunsen screen is used, this is the case when the semi- opaque and semi-transparent parts of the screen appear of equal brightness, and if the Lummer-Brodhun instrument^ is used, when the inner and outer parts of the field of view of the telescope through which the screen is viewed appear of equal brightness. If it is not possible to produce equality of bright- ness without moving the candles inconveniently near to or far from the photometer, increase or decrease the distance of the gas flame. Blow out the candles, weigh and replace them, and at a given instant relight them. Move them till equality of brightness of the screen is produced, and determine the distance of the candles from the screen by reading the scale position of a point vertically under the candles. If the photometer screen can be reversed this should be done and equality of illumination again secured by moving the candles. Their position should again be read on the scale. These observations should be repeated three or four times and the means taken. Rotate the gas flame so that the other flat surface is now presented to the photometer, and repeat the observations, taking the means. ^ This is the instrument shown in Fig. 94. XLIV PHOTOMETRY 215 In the same way take observations when the edges of the flame are presented to the photometer. Blow out the candles, noting the time, and again weigh them. Record as follows: — 19 Jan., 1897. Tulip gas burner. Distance D of gas flame from photometer screen = 80 cms. Weight of candles when lighted at lOh. 40m. 154*2 grams. „ blownoutatllh.lOm. 1460 .. Sperm consumed in '5 hrs. ( 8-2 gms. 127 127 grains. .*. Candle power of candles = X= ^ ' ^ = 2*12 standard ^ 5 X 120 candles. Gas flame Mean distances of candles from screen=<i (? D» D*ld* -? Candle power Flat (1) { 40cma 38 „ 1600 1524 6400 40 4-2 8-48 8-91 18-69 .. (2) { ' ■ Edge(l) { Ac. } .. (2) j I SECTION XLV. INTERFERENCE OF LIGHT. Apparatus required : Bunsen flame, sodium bead, slit, biprism, micrometer microscope, metre scale. When two beams of homogeneous light coming from the same source cross each other after having described paths differing in length, the vibrations due to the two may be in opposite directions and neutralise each other at certain points of the region where the beams cross. At such points the joint action of the two beams will produce partial or total darkness, and if a screen is placed in this region, a series of light and dark " interference " bands will be seen on it. In the case in which the source of light is a narrow vertical slit behind which a sodium flame is placed, and the two beams are produced by the passage of the light from this slit through a double prism the section of which is indicated in Fig. 95, the distance x between consecutive dark bands on the screen, is related to the wave-length X of the light, the distance a of the slit from the screen, and the distance apart c of the two virtual images of the slit Fig. 95. formed by the biprism, by the equation : X = -— . This method of producing interference bands may therefore be used to determine approximately the wave-length of light. The experiment is performed on an " optical bench," a simple form of which is shewn in Fig. 96. The slit consists of two brass plates with straight bevelled edges, attached by screws to a wooden stand provided with levelling screws. XLV INTERFERENCE OF LIGHT 217 The brass plates should be adjusted so that their edges are about 1 mm. apart and parallel to each other. This is most Fig. 96. easily done by placing a strip of thin writing paper between the edges, pressing the plates together and screwing them firmly to the wood, and then removing the paper. The biprism is mounted on a similar stand. The micrometer microscope consists of an eyepiece mag- nifying a few times, across the focal plane of which a cross wire, or by preference a pointed vertical needle, can be moved by means of a screw with a graduated head on which the number of turns can be read. The arrangement is mounted on a heavy wooden base. The three stands described above slide along a wooden bench provided with a groove in which two levelling screws of each stand slide, and with a gniduated scale. Place the slit vertical at one end of the stand, and the sodium Hame behind it, about 15 cms. in front place the biprism, and about 45 cms. in front of the biprism the micrometer eye- piece. If the slit, edge of biprism, and centre of field of view of the eyepiece, are in a line, and the edge of the prism is neixrly pamllel to the slit, a series of light and dark vertical bands should be seen on looking through the eyepiece. If no bands are visible move the biprism backwards and forwards along the bench and tilt it slightly in its own plane till bands are visible. Now watch the bands and continue the tilting by means of the single screw at one side of the biprism stand. Stop the tilting when the bands are most distinct. The edge 218 LIGHT XLV of the biprism will now be parallel to the slit. The biprism and eyepiece should now be moved along the bench till the bands are both distinct and at a convenient distance apart for measurement. When this is the case adjust the reference mark in the focal plane of the eyepiece to the centre of a light band, read the screw head, move to the next band, read, and repeat the process for 5 bands to the left of the centre, then omit 5 bands, and read the positions of the next 5. Place a metre scale over the slit, biprism, and eyepiece stands, and determine their distances apart. Record as follows : — 2 May, 1900. Sodium Flame. Band Beading Band Beading 1 2 3 4 5 37-72 38-12 38-54 38-84 39-25 11 12 13 14 15 41-48 41-85 42-20 42-59 42-93 Mean 38-494 42-210 Difference 1 3-716 Difference for 10 bands 1 scale division .-. Distance apart of bands = 3-716 scale divisions. = '05 cms. = •0186 cms. To determine the distance apart of the two virtual images of the slit formed by the biprism, place a lens of about 7 cms. focal length, provided with a stop of about '5 cm. diameter, in such a position between the biprism and eyepiece, that the two images of the slit are in focus. Measure the distance of the images apart by the micrometer, and the distances of the lens from slit and eyepiece by the metre scale. If a second position of the lens can be found for which the images are again in focus repeat the observations. XLV INTERFERENCE OF LIGHT 219 The virtual images, when formed b}' prisms of small re- fracting angles for nearly normal incidence, are at the same distance from the prism as the object. If therefore u is the distance of the slit from the lens, it will also represent the distance of the virtual images from the lens. The measured distance between the real images in the focal plane of the eye- piece being c,, the distance between the virtual images will be c u given by c = -^ , when v is the distance between the lens and the focal plane of the eyepiece. For a second position of the lens, considerations of elementary optics shew that u and v will be interchanged. Either of these observations will give c, but the measurement of u and v may be avoided altogether, for if we multiply the equations together we have c = \UiC^. Arrange observations and work as follows : — Distance of lens from Distance of images apart 8Ut = tt screen =r Scale divisions cms. c 19-7 cm. 40-2 403 cm. 19-8 780 193 •390 •0915 •191 •189 •189 Mean •190 u-\-v = 60*0 cms. = a. xc 0186 X 190 •'•^''-a" 60 •000,0580 = 5-80 x lO"" cms. Known wave-length of sodium light = 5'893 x 10~' cms. SECTION XLVI. MEASUREMENT OF THE WAVE-LENGTH OF LIGHT BY THE DIFFRACTION GRATING. Apparatus required : Spectrometer, diffraction grating, Butisen flame, and sodium head. When a beam of light diverging from a point passes through a transparent plate on one surface of which a series of equidistant fine parallel lines has been ruled, the emergent light appears to diverge from a number of virtual images, one of which is coincident with the luminous point, the others, which lie on either side of the luminous point, are called diffraction images. The surface on which the lines are ruled is called a diffraction surface or grating. The connection between the relative positions of the vat-ious images, the distance apart of the lines, and the wave-length of the light transmitted, can be calculated, and experimental observation will therefore furnish a means of measuring the wave-length of light. In the case of a thin parallel plate of a transparent medium with equidistant parallel lines ruled on one surface, if the light incident on the ruled surface is a parallel beam, each diffracted beam will also be a parallel one, and will be deviated by the angle 6, where n\ = b sin 6, n being an integer called the order of the image, and h the distance of the lines apart measured from centre to centre. Hence if the grating is placed on the table of a spectrometer, the collimator of which furnishes a beam of parallel light, and the telescope of which is focussed for parallel rays, the whole of the images will in turn be visible through XL VI THE DIFFRACTION GRATING 221 the telescope as it is moved from one side to the other. The deviation of each diffracted beam may therefore be deter- minated on the graduated circular scale of the instrument. If a narrow illuminated slit is used instead of a luminous point, the images will be linear. Adjust the spectrometer as described in section XLI. Place a sodium flame behind the slit. Using the Gauss eyepiece in the telescope direct it towards the collimator, adjust the centre of the cross wires to the image of the slit,, and clamp the telescope. Place the diffraction grating on the table of the instrument with the plane of the ruling perpendicular to the line joining two of the screws of the table. Rotate the table till the image of the cross wires formed by reflection at the surface of the gmting is seen in the field of view. Adjust the table by means of the levelling screws till the reflected and direct images of the wires coincide. The gloating is then perj^endicular to the axis of the collimator and telescoj>e. Remove the Gauss and substitute an ordinary eyepiece. Rotate the telescope till the first diffracted image is on the cross wires. If it is indistinct the screw of the table which will tilt the lines of the grating in their own plane, should be rotated till the image is as distinct as possible. The width of the slit should then be varied till the two sodium lines are resolved quite distinctly. Adjust the telescope by means of the tangent screws till the centre of the cross wires coincides with one of the lines, and read the vernier of the circular scale. Adjust to the other line and again read. Now rotate the telescope, adjust to the direct image and read the circular scale. Then rotate further in the same direction till the first diffi-acted image on the other side is .seen, adjust and reiid the scale for each line. Again return to the direct image and take rciulings. By rotating still further in each direction, the 2nd, 3rd, &c. images may be found and their positions read. To obtain the wave-length of light the distance b between the lines must be known. In an absolute determination the 222 LIGHT XLVI most difficult part of the measurement consists in finding this distance. It may be determined by means of a microscope, but in the present exercise b will be assumed to be known. Record the observations as follows : — 6 Jan., 1899. Spectrometer A. Grating A. Most refrangible Sodium line. 1 Vernier A Vernier B First order, right 202° 11' 30" 22° ir 00" Direct reading 179 10 30 359 10 00 First order, left 156 9 30 336 9 30 Deviation, right 23 1 00 23 1 00 left 23 1 00 23 30 „ mean 23 1 00 23 45 Mean deviation for first order spectra ... 23° 0' 52". Similarly for the other line, and for the spectra of the 2nd and higher orders. Tabulate results as follows : — Grating space of A 6 Jan., 1899. = 6 =00015062 cms. n Angle of deviation = d sin e sin ^ 1 sin 6 n n 1 2 23° 0'52" 51 25 •3909 •7817 •3909 •3908 •000,05888 •000,05887 Mean •000,05888 If the telescope is now turned to one of the first order images, and the table on which the grating is placed is slowly rotated, it will be seen that the position of the image, and therefore the deviation, change. A position of the grating can be found for which the deviation is a minimum. Adjust the telescope on the image when least deviated, and read the XLVI THE DIFFRACTION GRATING 223 veraiers. Keeping the table fixed, rotate the telescope till the direct image is seen. Read the verniers. Now rotate table and telescope till the minimum position of the first image on the other side is found, and repeat the observation of the deviated and of the direct image. The minimum deviation </> is connected with X and b by the equation 2b sin ^ = nX, hence X may again be determined. Tabulate as follows : — First order spectra. Vernier A. 202° 42' 1 80 8 30 = 22° 31' 30 Vernier B. 22^^ 42' 30' 10 22° 32' 30' 22° 32' 0" Deviated reading right Direct „ Deviation to right Mean Deviated reading left ... = Direct „ ... = &c. Deviation to left ... = Mean ... ... = Mean deviation for first order spectra = 22° 32' 15". Similarly for the second order spectra, collecting the results in tabular form as follows : — 6 January, 1899. n 2 .»! ^-1 f-l- 1 2 22- 32' 15" 46 2 ir 16' 7" 23 1 •1954 •3910 •1954 •1955 •000,05887 5889 Mean •000,6888 By using the grating so that the diffraction images are formed by reflected rays, further determinations of X may be made. SECTION XLVII. ROTATION OF PLANE OF POLARISATION. I. Apparatus required : Two NicoVs prisms, two tubes with glass ends, a solution of sugar, and a sodium flame. When a beam of plane polarised light is transmitted through certain solids, liquids, and solutions, it is found that the plane of polarisation is rotated through an angle pro- portional to the length of the path of the ray in the substance. The object of the present section is to verify the laws of rotation of the plane of polarisation of light by solutions of certain substances. In these cases the rotation is very nearly proportional to the mass of dissolved substance per c.c. of the solution. The rotation produced by a layer 1 cm. thick of a solution containing 1 gram per c.c. is called the specific rotatory power of the substance. If a solution contains in 1 c.c. a grams of a substance the specific rotatory power of which is p, and a beam of plane polarised light passes through I cms. of the solution, and R is the rotation of the plane of polarisation produced, then R = a.l .p. Take about 85 c.c. of water, and add to it 10 grams of the sample of sugar provided. When the sugar has dissolved dilute the solution till its volume = 100 c.c. If the solution is coloured, mix with it about 2 grams of bone black, allow it to stand a few minutes and then filter. If the filtrate is then bright and clear it may be used. Arrange the solution tube provided, and two mounted Nicol's prisms, in such a way that the light of a Bunsen XLVII ROTATION OF PLANE OF POLARISATION. I 225 flame containing a bead of a sodium salt can be seen through the tube and prisms. The Nicol without the circular scale is to be placed between Fig. 97. the flame and one end of the tube, i.e. used as the " polariser," and that with the index and circular scale graduated in degrees, [^M Fig. 98. placed between the other end of the tube and the eye, i.e, used as the " analyser," Figs. 97 and 98. A thin sheet of yellow glass may be placed between the flame and polariser to cut off the blue light of the flame, and if necessary^ a lens may be used between the analyser and the eye. Wash out the tube thoroughly, and after filling it with distilled water place it between the prisms and rotate the s. p. 15 226 LIGHT XLVII analyser till no light passes through the system. This will be the case when two lines similarly placed in the Nicols and at right angles to the line of sight are also at right angles to each other. Read the circular scale on the analyser. Repeat the observation several times, approaching the point of extinction from opposite sides each time, and take a mean of the results. Dry the tube by passing through it a plug of cotton-wool, and then fill it with the sugar solution, place it between the prisms, and determine as before the position of the analyser when no light passes. Repeat, and take the mean. The difference between the two means is the rotation produced by the solution. To test the truth of the law expressed by the equation R = a.l.p, take 20 c.c. of the solution, and dilute to 40 c.c. Fill a tube with this solution, and determine the rotation pro- duced. It should be half the amount previously obtained. Wash out another tube, fill with the solution, and place it between the prisms so that the light now passes through the two tubes in succession. Determine the reading for extinction. The rotation produced by both tubes will be found to be double that produced by one, and equal to that produced by one tube of double the strength. Arrange your results as follows : — 17 Oct., 1895. Reading for darkness, with water tube ... ... 167 '2° „ „ with original solution 156*4° Rotation for original solution ... ... = 10*8° Reading for darkness, with solution of half strength = 161 '7" Rotation for solution of half strength ... = 5*5'' Reading for darkness, with 2 tubes of solution of half strength = 156*r Rotation for double length of solution of half strength ... ... ... ... ... = 11*1^ SECTION XLVIIL ROTATION OF PLANE OF POLARISATION. II. Apparatus required: Laurent or other polamineter, sugar, and sodium jlaine. With the apparatus used in the previous exercise the eye has been called upon to judge the point at which the minimum light passed through the Nicol's prisms. As the estimation of the exact position for a minimum is difficult, several instru- ments have been devised for getting over the difficulty, one of these being the Laurent polarimeter provided. By their means it is possible to estimate the position of extinction to a small fraction of a degi-ee. The Laurent instrument is arranged as shewn in Figs. 99, 100. ttt^ Fig. 99. The light from the Bunsen burner A, in which a bead of a mixture of equal parts of sodium diborate and common salt is placed, passes through the lens li, the small hole in the diaphragm C, which is covered by a thin plate of bichromate of potash to cut off all but the yellow light, and falls on the lens and Nicol prism D. The plane polarised light which emerges from the Nicol falls on a plate of quartz E, which covers half the field. The quartz plate is cut so that the 15—2 228 LIGHT XLVIII optical axis is parallel to the edge which bisects the field. The plane polarised light falling on the plate is decomposed into two rays, one polarised in a plane parallel to the edge of the plate, the other in a plane perpendicular to this edge. The two rays traverse the plate with different velocities, and the thickness of the plate is so arranged that a difference of phase of half a wave-length is produced. The effect of this is, that if the light passing through the uncovered half of the field is polarised, say in the direction CA inclined at an , angle 6 to CB the edge of the quartz plate, then that which has passed through the plate is polarised in the direction CA' such that BCA' = BOA. On looking through the analysing Nicol K of the eyepiece, the two halves of the field will appear unequally bright, unless the principal plane of the analysing Nicol makes equal ^^' angles with the directions GA, GA\ i.e., is either parallel or perpendicular to GB. If it is parallel to GB the halves are equally bright, if perpendicular equally dark. The dark position is the one made use of, and the instrument is more sensitive the smaller 6, is consistently with sufficient light passing through the apparatus. Generally 6 does not exceed 2°. Adjust the eyepiece till the dividing line between the fields is seen distinctly when one half of the field is dark and the other light. The Nicol D is connected to a horizontal moveable arm, shewn in Fig. 100, and may be rotated within certain limits. Id order to get the position of greatest sensitiveness, determine first the position of the arm when the two halves of the field are equal in brightness whatever be the position of the Nicol K. This should be done with an empty tube at F. Now rotate D through a small angle not exceeding 2° by means of the moveable arm. It will then be found that the two halves of the field are equally dark for a certain position of K, the left- hand half increasing, the other decreasing, in brightness if K is moved in one direction, and the right-hand half increasing, the other decreasing, if the rotation is in the opposite direction. By means of the rotating screw (r, adjust the vernier to read 0, and by means of the tangent screw H, rotate K till XLVIII ROTATION OF PLANE OF POLARISATION. II 229 equality of fields is again produced. The instrument now reads when no active material is present. Insert at jF a tube filled with distilled water, and determine Fig. 100. the reading for equality of the fields, the adjustment being made alternately from opposite sides of the position of equality. This reading should still be if the water were pure, and the glass ends of the tube unstrained. Now insert a tube filled with a sugar solution containing 10 grams of sugar per 100 c.c, and take readings several times as before. The sample of sugar from which the solution has been derived" may contain both sucrose or cane sugar, and glucose 230 . LIGHT XLVIII or invert sugar, and impurities which will be assumed to be inactive. The cane sugar rotates the plane of polarisation to the right, and the invert sugar to the left. The observed rotation will be due to the difference of these effects. To determine the amount of each constituent present, we make use of the fact that when cane sugar is heated gently with acid it is converted into invert sugar, so that the whole of the sugar then present in the solution is invert sugar. From the two observations of the rotation the amount of each constituent present can be calculated. Take 20 c.c. of the original solution of sugar in a flask, add to it 2 drops of strong hydrochloric acid, add water till the volume is 22 c.c, and warm gently for 10 minutes, keeping the temperature about 80° C. Cool the resulting invert sugar solution by placing the flask in water, and after its temperature has fallen below 20° C. note the temperature, insert the solution in the tube, and determine the rotation produced. Let Ri be the rotation observed with the original sugar solution, and let R^ be that observed with the invert sugar, rotation to the right being considered positive and to the left negative. Let pi be the specific rotatory power of cane, p2 that of invert sugar, and let a^ grams of cane sugar and a., grams of invert sugar be present in 1 gram of the sample. Then in the first experiment, T^ 10a, -_ 10a., ^^ = 2/3iai + 2p2 ^2 (!)» and in the second experiment, J. _ JlOa, 360 lOaJ 20 ^"^^tl00'342^100J 22"^^' The factor 360/342 being due to the change from CiaHasC^n to C12H24O12 in inversion of the sugar, and the factor 20/22 due to dilution with acid. Reducing we have l'lR, = 2pAai-^-\-a} (2). XLVIII ROTATION OF PLANE OF POLARISATION. II 231 Subtracting (2) from (1) we have R, - IIR, = {2p, - 2105/33) a,. Hence a, = ^ — _- __ , and a.^ = - - = — „ — / T^* 2p, - 2-105/5, ' p, 2p^{px-p2) The quantity pi has been found to be nearly independent of temperature. Its value is + 6'65 degi-ees. /3o is negative and depends on the temperature t. Its value is - (2-78 - -03210 degrees. Record observations and results as follows : — 15 Jan. 1896. Solution containing 10 grams of sample per 100 c.c. Reading with water in tube ... O'lO' to right. Do. sugar solution ) og^, 12^24', 12^30', 12°30', 12^^24' ) "'^^''' ^iL. " " Rotation at temp. 15° C 12^7' „ „ Readine: with invert solution ) ^oi/x/ . i r^. 3° 5(y, 4^30'. 4°5', 4^5' } "^^^°' _^ '" ^^^'• Rotation at 14° 5' C 4° 20^ „ „ Value of p2 M » ••• ••• =— 2°-31 i2, = 12°-28, 122 = -4"-33. _ 12-28- 11 (- 4-33 ) _ 1714 Hence a, - ^^.^^ _ g.^^. ^_ ^gi) - 18-15 " ^*' 614-6-25 ^. "^= -2-3r-*^"- Or the sample contains 94% of cane sugar and 5^0 of invert sugar. Wash out the tubes thoroughly with tap water before putting away the apparatus. BOOK VI. MAGNETISM AND ELECTEICITY. SECTION XLIX. THE MEASUREMENT OF MAGNETIC FIELDS. Apparatus required : Mirror magnetometer, lamp and scale, bar magnet, and vibration box. The intensity of a magnetic field at a point is measured numerically by the force on a unit pole placed at that point. It can therefore be represented by a straight line, the direction of the line representing the direction, and the length representing the intensity, of the force. If two fields are superposed on each other the intensity of the combined field may be obtained in the same way as the resultant of two forces. If OP, Fig. 101 a, represents the direction and magnitude of the magnetic force on a unit pole placed at 0, due to one of the fields, and OQ that due to the other, the combined effect will be represented by the diagonal OR of the parallelogram OPQR. The magni- tude and direction of the resultant field may be calculated like the corresponding quantities in the case of a resultant force, by the following equations: OR' =^ OF' +0Q' + 20P.0Q cos POQ (1), OP ^ OQ ^ OR ain QOR sin POR sin POQ ^ ^' XLIX THE MEASUREMENT OF MAGNETIC FIELDS 233 Two special cases are of frequent occurrence. If OP and OQ are at right angles to each other (Fig. 101 6), the magnitude of the resultant is given by OR^ =0P'-\-0(^ and tan QOiJ=^ (3). Fig. 101 o. Fig. 1016. The second case (Fig. 101 c) is that in which the resultant OR is at right angles to one of the forces OP. Then OR* = OQ* - OP*, and sin Q0ie = 5|. Exercise. — It is found that the magnetic force on unit pole at a point distant r cms. from a straight wire of infinite length through which a current of strength i units is passing, is equal to 2i7r, and the lines of force are circles having the wire as axis. If the wire is vertical and carries a current of one unit, find by calculation or geometrical construction the direction in which a small magnetic needle will point when placed respectively east, west, north, south, and north-east of the wire at distances of 4, 11, or 15 centimetres from the wire. The earth's hori- zontal magnetic force may be taken as '17. If the magnetic field is due to a magnet having its north pole of strength fi Q,t B and its south pole of strength — fi at A, the magnetic force on unit pole at a point P due to the north pole at B is fi/BP', that due to the south pole at A is — /it/ilP*. If the point P is on the straight line through the poles A and B, the two forces act in the same straight line, and have a 234 MAGNETISM AND ELECTRICITY XLIX resultant in the direction of the stronger, of intensity / such that T- ^ J_ _ 1_^- AB{BF-vAF) '^\BP' AP") ^ BP'AP' If we call the magnetic moment fjuAB of the magnet M, the distance of P from the centre of AB d, and if I is half the distance between the poles, i.e. about -f^ the length of the magnet, the resultant intensity 2Md ,. 2d = M ^d-iy{d + iy {d^-pf If the point P is so far from the magnet that I is small 2 compared to d, we have simply I = M.-^^. If the axis of the magnet is at right angles to the magnetic meridian, and H is the earth's horizontal force at the point P, the angle which the resultant force makes with the magnetic meridian is such that tan ^ = Tr > £1 H- Yd—'^''^ W- A small magnetic needle placed at the point at which the resultant has been calculated will set itself in a direction forming an angle 6 with the magnetic meridian, and therefore enable 6 to be found. The resultant magnetic force at a point P due to a magnet AB, which in the above case has been calculated when P is on AB produced, may also be easily determined when P lies on a line through the centre C of the magnet perpendicular to AB. Since AP = BP the magnetic forces due to the two poles are equal, and their resultant bisects the angle between their directions, i.e. is parallel to AB. Each force having an in- tensity -jy^ has a component . ^g parallel to AB, hence the mtensity /= ^pa • Writing again M for the magnetic moment of the magnet, XLIX THE MEASUREMENT OF MAGNETIC FIELDS 235 / for half the distance between the poles, and d for the distance of P from C, we have I ~ r^ ;,T| • If P is so far from C that M I is small compared to rf, this becomes / = ^ . Hence the field at a distance d from the centre of a short magnet along the axis of the magnet, is twice the field at the same distance along a line through the centre of the magnet perpendicular to the axis. If the law of force had been the inverse ?jth power instead of the inverse square, the former force would have been found to be n times the latter. Hence an experimental determination of the ratio of these forces will furnish a proof of the law of action of magnetic poles on each other. If the axis of the magnet is at right angles to the magnetic meridian, and H is again the earth's horizontal force, the resultant of the two fields will make an angle 6 with the meridian where tan 6 = ^. or ^=((^-^ + Z-')Uan6> (5). The angle 6 may as before be determined by ])lacing a small magnetic needle at the point P, and observing the deflection produced when the magnet is placed in position. The apparatus provided consists of a small magnetic needle to which a mirror is attached, suspended by a fine fibre the torsion of which may be neglected. The centre of the needle is situated vertically over the middle points of the two horizontal graduated scales placed at right angles to each other, one of them being in the magnetic meridian. For con- venience the latter should be about 2 mms. below the former. The rotation of the mirror is determined in the usual manner by the motion of the image of a cross wire formed after reflec- tion by the mirror, on a scale placed parallel to the mirror when in its central position. The cross wire is attached to the scale and illuminated by a lamp. (For the method of obtaining the angles of rotation see Section XXXIV.) 236 MAGNETISM AND ELECTRICITY XLIX The deflecting magnet provided is first placed on the scale running east and west, say to the west of the needle, with its north pole towards the needle, the positions of the ends of the magnet are observed, and the scale reading of the cross wire is taken. The magnet is then reversed so that its south pole now points towards the needle, and the deflection again ob- served. It is then transferred to the east of the needle and the two deflections again observed. To make the corresponding observations with the magnet north and south of the needle, the magnet is placed on a frame which slides along the scale in the magnetic meridian, the magnet itself being at right angles to the meridian. The height of the frame is such that the magnet itself will be at the same level as when placed on the other scale. The frame is moved along the scale till the centre of the magnetic axis is at the same distance from the centre of the needle as pre- viously, the north pole of the magnet being say to the east. The deflection of the cross wire is observed, the magnet is reversed so that its north pole points to the west, and the obser- vation repeated. The frame and magnet are then transferred to the other side of the needle and the observations repeated. A set of observations should be taken for each of three distances of the magnet from the mirror as far apart as possible, and each set recorded as follows : — 18 March, 1895. Determination of MjH at station B. Deflecting magnet A, length = 6"0 cms. .'. Z = 2*5 cms. Magnetometer 5, distance of scale from mirror = 103 cms. Position of Magnet Magnetometer mean tan 2d tand N. end cms. 8. end cms. Middle cms. Beading cms. Deflection cms. P( 250 W 31-0 W 31 -OE 25 E )sition of r< 310 W 250 W 250 E 31-0 E 3St 28-0 W 28-0 W 28-0 E 28-0 E 500 68-5 31-6 69-3 30-2 zero 18-5 18-4 19-3 19-8 19-0 •184 •092 XLIX THE MEASUREMENT OF MAGNETIC FIELDS 237 Similarly for the north and south positions. For the E. and W. positions in the above example we have:- M H 2d ob 996. Similarly for the other positions. In order to determine both M and H it is necessary to find the value of some combination of the two other than the quotient, and the most convenient combination to detemiine experimentally is the product MHy which is connected with the time of torsional oscillation r of the magnet about a vertical axis through its centre of gravity, by the equation 27r y MB where / is the moment of inertia of the magnet about the axis of oscillation. 47r2/ Hence MH^ To determine the product MH, place the magnet which has been used as the deflecting magnet, in a light stirrup supported by a long thin fibre the toi-sion of which can be neglected, so that it can oscillate in a horizontal plane about a vertical axis. To protect the magnet from air currents suspend it in a box. Observe the times at which one end of the magnet passes a fixed mark on the bottom of the box in one direction, for six consecutive passages, the arcs of vibration not exceeding 20*". Wait a time nearly equal to that between the first and last observations, and then take six more observations. Arrange as follows : — Passage No. First Set ' Passage No7 Second Set Time of 10 oscillations Ih. 3m. 28. 10 Ih. 4m. lis. 69 8008. 1 9 11 17 68 „ 2 16 12 24 68 „ 3 23 13 30 67 ,. 4 29 14 38 69 „ 5 36 ; 16 43 67 „ Mean Time of one oscillation t »68 = 6-8 MAGNETISM AND ELECTRICITY XLIX Determine the moment of inertia of the bar magnet by weighing it to '01 gram, and measuring its length and breadth. The moment of inertia /, if m is the mass of the magnet, 2a its length, 26 its breadth, is given by the equation In order to determine whether a magnet is weak or strong, it is advisable also to calculate the magnetic moment per unit mass. Arrange observations and calculations as follows : — Mass of magnet ll'Ol grms. a= 3-0 cms. a2= 9-0 h= -23 „ 6^= -05 a^ + b^= 905 .-. 7 = 1101 X 302 = 33-2 , ,,^ 4 X 9-87 X 33-2 „„ . and MH = -r^-^ = 284. M Having obtained -^ and MH, M and H are determined as follows : — M = \/{^ {MH) = V996 X 28-4 = V2'8300 = 168. -1 £»0 Magnetic moment of magnet per gram = ,- = 15*2. SECTION L. MAGNETIC SURVEY OF THE LABORATORY. Apparatus required : Magnet, vibration boXy and watch. If magnetic observations are taken in a room not specially constructed for magnetic work, it is quite possible that iron beams or iron pipes about the room may have an appreciable influence on the results. It becomes advisable under these circumstances to make a magnetic survey of the room, that is to determine the intensity and direction of the earth's magnetic force at a considerable number of points in it. As change of direction of the horizontal magnetic force is in most cases less objectionable than change of intensity, attention should be given chiefly to the latter. It will not be necessary to measure the horizontal force at different places, but only to compare them with each other. This is done in the simplest way by measuring the times of vibration of a magnetic needle ab<jut a vertical axis of suspension, as the horizontal forces will be inversely proportional to the squares of these times. The apparatus supplied is provided with a horizontal scale of degrees which enables the direction of the magnetic force, i.e. of the magnetic meridian, to be detennined. A magnetic needle to which a pointer moving over the scale is attached, is sus- pended from a rod by means of a fibre which should be free from torsion. To secure this, take out the needle and suspending fibre, and holding the needle in the hand allow the suspending rod to hang down until the fibre is completely untwisted. The direction and magnitude of the earth's magnetic force are to be determined at each of the places shewn in the plan of the laboratory*. * A plan of the laboratory with the positions at which observationn are to be taken marked on it, ihoald be taapended in the laboratory. 240 MAGNETISM AND ELECTRICITY At each place the box within which the magnet swings, is to be placed with its long sides, and therefore the zero line of the scale, parallel to the outer wall of the laboratory. The needle should then be set into oscillation about the axis of suspension, and the readings of one end at five successive turning points, three on one side of the position of rest and two on the other, taken. The mean of the three on one side, and the mean of the two on the other, should then be found, and the mean of these means taken. Repeat, reading now the other end of the needle. The mean of the results for the two ends is the angle between the magnetic meridian and the outer wall of the laboratory. To measure the time of oscillation with sufficient accuracy for our purpose, note down to the nearest second as in the example given below, the times at which the needle passes through its position of equilibrium in the same direction, during say six successive swings not greater than 20°. Then wait a time approximately equal to that between the first and last observations and take another set of six observa- tions of the times of passage. Two columns may now be formed by entering the observed times as follows : — OsciUation No. First Set Oscillation No. Second Set Time of 10 oscillations 1 2 3 4 5 11 h. 23 m. 3 sec. ' 13 „ ; 17 „ I 21 „ i 25 „ 1 10 11 12 13 14 15 23 m. 49 sec. 53 „ 59 „ 24 m. 4 „ 9 » 13 „ 46 sees. 45 „ 46 „ 47 „ 48 „ 48 „ Mean = 46-7 „ Time of 1 oscillatioD = 4*67 „ The differences in time between corresponding passages in the first and second sets are entered in the third column and the mean taken. This mean should be correct to half a second, i.e. the error should not exceed 1 per cent. MAGNETIC SURVEY OF THE LABORATORY 241 In order to make certain that no mistake in the number of oscillations has been made, the experiment should now be repeated. If during your experiments there are any movable masses of iron within a metre of the magnetic needle, record the fact and state their positions. The horizontal force being inversely proportional to the square of the time of oscillation, we may obtain numbers pro- portional to the horizontal force by calculating the reciprocals of the squares of those times. (Barlow's Tables will save much arithmetical work.) Taking the given values of H and of the magnetic declina- tion for the point of reference marked in the plan, determine the intensity and direction at the other points. Enter results as follows, giving also the plan with the points of observation marked on it. 26 March, 1900. 1 Angle to Angle to Place of observation T fi H Laboratory Walls 19° E Geographical Meridian Remarks 1 11-61 •00741 -192 3°W 2 10-61 885 -221 8"E 14° W^ 3 14-61 467 -126 69° E 47° E 1 much 4 8-77 -01299 -323 21° E rw disturbed 6 10-81 855 •221 9°E 13° W J 6 12-38 653 •168 8°E 14°W 7 12-27 662 •172 3° W 25° W 8 12-33 •00657 •170 5°E 17° W reference point ■.P. 16 SECTION LI. THE TANGENT GALVANOMETER AND OTHER CURRENT METERS. If a wire is bent into the form of a circle of radius r, and a current of strength i c.g.s. units is sent through it, a magnetic field is produced, which, at the centre of the circle, and at distances from the centre not greater than '08 of the radius, is sensibly uniform, of strength F = ^irilr, and in direction perpendicular to the plane of the wire. Taking the earth's magnetic field into consideration, the total field at the centre of the circle is the resultant of the fields due to the earth and the current in the wire respectively, and a magnetic needle supported at its centre will set itself along the direction of the resultant field. If the plane of the wire is coincident with the magnetic meridian, the component forces are at right angles, and if 6 be the inclination of the resultant to the magnetic meridian, and H the earth's horizontal force, we have tan = -rj. Hence, substituting the value of F in terms of i, we have rH i= TT- tan 0, ZTT which will enable us to determine i ii H and r are known, and 6 is observed. From the presence of the tangent in this expression, this arrangement of a circular coil of wire with a magnetic needle at its centre is called a Tangent Galvanometer. LI CURRENT METERS 243 In order to make the instrument more sensitive, the current to be measured may be sent through a wire coiled n times round the ueedle. In that case the magnetic field produced by the current is n times that produced by a single wire of radius equal to the mean of the radii of the n turns, and we have *=£*-^ w- The practical unit in which currents are measured is the Ampere, which is one-tenth of a c.g.s. unit, hence, if we call A the number of amperes passing through the coil, we have ^ = ^l^tan(9 (2). riTT Since 5r/mr depends only on the coils of the galvanometer, we may put it equal to l/G and call G the " coil constant of the galvanometer," and we have A = ^ tan 0. (jr If the galvanometer is used in a fixed position, H will be constant, and we may write H/G = k. We then have A =k tan 0, where k is the current which produces a deflection of 45°, and is generally called the " current constant of the galvanometer." It is, however, unsafe to assume that H is uniform through- out buildings, owing to the irregularity of the distribution of the earth's line of force, produced by masses of iron in the walls and floors. Hence, if a current is to be measured in amperes by means of a tangent galvanometer, the horizontal component of the earth's magnetic field at the point of observation must be determined. This is best done by taking the times of oscil- lation of a magnet about a vertical axis, at the point of observation, and at some point at which the value of H is known. (See p. 241 Magnetic Survey, or Intermediate Practical Physics, p. 184.) In order to form a judgment of the power of the instru- ment, we must estimate the accuracy with which an observation 16—2 244 MAGNETISM AND ELECTRICITY LI of the deflection may be taken, and calculate from this the en-or which may be introduced in the result. If the circle of the galvanometer is properly graduated, we ought to be able to read the deflections to '1 degree, but it would be difficult to secure a much greater accuracy. An error of "1 division will according to the investigation (p. 6) affect the determination of the current much more in some parts of the scale than in others. Numerical Exercise. Calculate the percentage error introduced in measuring a current by an error of '1 degree in the reading of a tangent galvanometer, the deflections being 10°, 20°, 30°, and so on to 80°. When the deflection of a galvanometer needle changes, its poles trace out a circle, and in order that the tangent law may be accurate, the magnetic forces F and H should be the same at all points of the circumference of this circle. We require, therefore, to know how long the magnetic needle may be without the deviations from the tangent law introduced by variations of F exceeding the errors introduced in other ways. It may be shewn that the greatest error due to the finite length of the needle is introduced when the deflection is 60°, that it is proportional to the square of the length, and that if the ratio of the length of the needle to the diameter of the coil is less than 08, the error introduced by the want of uniformity of the field is never greater than that caused by an error of '1° in the reading of the deflection. Equation (2) shews that the sensitiveness of a tangent galvanometer may be increased either by diminishing the radius r, or by diminishing H, or by increasing the number of turns n. If the radius of the coil is too much diminished, the tangent law ceases to hold unless the needle is also shortened. The horizontal force H may be diminished and the sensi- tiveness in consequence increased by permanent magnets placed .80 as to counteract the earth's magnetic force. Numerical Exercise. Design a tangent galvanometer of -one turn which, when placed in a field of strength '17, will give a deflection of 45° for a current of 1 ampere. LI CURRENT METERS 245 Adjustment of the tangent galvanometer. A tangent galva- nometer before use should be adjusted so that the plane of the coil through which the current is to be sent, is parallel to the magnetic needle. The coil will then be in the magnetic meridian. As the deflected magnet is short, a pointer is attached to it, so that the rotations may be more easily read ofif on a circular scale. When the needle is in the plane of the galvanometer coil, the pointer should stand at the zero of the scale. As this condition may not be accurately satisfied, any error is eliminated by sending the current through the galvanometer first in one direction and then in the other, the deflections being noted each time. If the zero of the scale is correctly placed, the two deflections will be equal and opposite. If the adjustment is not quite right, one deflection will exceed the true deflection by the same amount that the other is in defect, and the mean of the two will give the true deflection. If the axis about which the needle rotates is not quite at the centre of the scale, small errors will be introduced which are, however, eliminated by reading the two ends of the pointer (see note p. 202). Each observation of a tangent galvanometer consists, therefore, of four readings, the mean of which gives as correct a value as is obtainable with the instrument. In a second type of current measuring instruments, known as electro-dynamometers, the magnetic needle of the galvano- meter is replaced by a coil of wire through which the current to be measured is sent. The passage of the current through the coil makes it behave as if it were a magnet with its axis along the axis of the coil. The coil will therefore set itself with its axis along the resultant field due to the earth and to the current in the fixed coil. In general the movable coil is suspended by means of the wires which carry the current to and from it, and the suspension introduces another force, tending to keep the coil in the position it occupies when no current passes, i.e. with its axis in the plane of the fixed coil. If the force introduced by the suspension greatly exceeds that due to the presence of the coil in the earth's field, the latter may be neglected, and a direct relation between the strength of the current in the two coils and the deflection obtained. 246 MAGNETISM AND ELECTRICITY LI Instead, however, of allowiug the movable coil to be deflected, we may increase the force due to the suspension, by twisting the head carrying the upper ends of the suspending wires in a direction contrary to the deflection, till the deflection is reduced to zero. The magnetic force tending to deflect the movable coil, is then equal and opposite to the force on the coil intro- duced by the rotation of the torsion head. The magnetic force is proportional to the magnetic moment of the suspended coil, and to the strength of the field in which it is placed. It will therefore be proportional to the square of the current, since both the strength of the field in which the suspended coil moves, and the moment of the magnet to which the suspended coil is equivalent, are proportional to the current. In the Siemens electro-dynamometer the torsional force is provided by a spiral spring, to the top of which a pointer moving over a dial is attached. This pointer is rotated by means of the torsion head till a second pointer attached to the moving coil is at the zero of the scale. The planes of the two coils are then at right angles to each other, and the torsional force exactly balances the magnetic force. The torsional force being proportional to the angle 6 of twist of the torsion head, we have C^ x 0, or C = K\/d, where K is the constant of the instrument. In order that the deflecting effect of the earth's field on the suspended coil may be least, that coil should be placed with its plane perpendicular to the magnetic meridian. Another form of instrument depending on the action of currents on currents, is the Kelvin current balance, where two horizontal coils are suspended from the arms of a balance, and each is acted on by two similar coils placed above and below it respectively. The current to be measured is sent through all the coils, in such directions that the suspended coil on the left is pulled down and that on the right pulled up. This pull is balanced by a weight sliding along a scale attached to the arm of the balance, and the distance through which this weight has to be moved to produce equilibrium, will be proportional to the square of the current passing through the coils. Another type of galvanometer which has recently come LI CURRENT METERS 247 much into use is the D'Arsonval. It consists of a movable coil suspended in a magnetic field, the lines of force of which are parallel to the plane of the coil. When a current is sent through the coil, it tends to set itself so that the greatest number of lines of force pass through it, and a deflection results which can be read off on a scale, a beam of light being reflected from a mirror fixed to the coil. The magnetic field of the instrument being many times stronger than the earth's field, its indications are independent of magnetic disturbances, such as occur in the neighbourhood of electric railways. If the current an instrument is intended to measure is very small, the coils consist of a great many turns of fine wire, and if the resistance of the instrument, which is in consequence high, is constant, the indications may be taken as measures of the electromotive forces or differences of potential applied to the terminals, i.e. the instrument may be used as a "voltmeter." SECTION LIL COMPARISON OF CURRENT METERS. Apparatus required : A small storage cell, a tangent galvanometer arranged so that one, two, or three turns of wire can he used, the constant with the three turns being about 1, a Siemens Dynamometer, having a constant about % a set of resistances each about 'o ohm which can be joined up in different ways so as to alter the resistance in circuit, and a commutator to enable the current to be reversed. Fig. 102. B \1M Si @ R K Fig. 102 a. When instruments intended for the same purpose are con- structed on different lines, it is advisable to compare their indications together, and the present exercise consists in com- paring the indications of the tangent galvanometer with those of an instrument such as the electro-dynamometer or Kelvin's " ampere balance," constructed according to a different principle. LI I COMPARISON OF CURRENT METERS 249 Connect the storage cell B (Figs. 102, 102 a) through the resistance coils R to two opposite quadrants of the four-way key Ky the plugs being removed. Place the electro-dynamo- meter S with the plane of its fixed coil in the meridian, at least 20 cms. from the resistance coils, and the tangent galva- nometer G with its coil also in the meridian, a metre away from the dynamometer. Connect them in series with each other to the other terminals of if as shewn. See that the galvanometer needle swings freely, then level the dynamometer, and screw down the spring on the base till the movable coil is released. Rotate the torsion head in the centre of the dial, and observe whether the coil moves freely. Arrange three of the '5 ohm resistance coils in series. Connect together the two wires leading to the tangent galva- nometer by placing both under the same binding screw on the base of the instrument, and make the circuit at K by inserting plugs in the two holes in one diameter of the key. While contact is being made, watch carefully the needle of the galvanometer in order to detect if there is any displacement due to a direct action of the current in the leads. If so, twist the leads between K, G, and S round each other. This probably will stop the action. If it does not, reverse the direction of the current by inserting the plugs in the holes in the other diameter of the key and see whether the effect is reversed ; if not, the cause is to be found in a direct effect of the current in the coils R on the galvanometer needle. Change the position of R till this effect disappears. All extra- neous effects of the current on the magnetic needle of G are thus got rid of before the galvanometer is used as a measuring instru- ment. In all future experiments students will be expected to avoid errors due to the magnetic effects of the leads on the measuring instruments, without being specially directed to do so. Connect the wires leading to the galvanometer to the two terminals of the instrument between which 1 T {i.e. one turn) is marked, and complete the circuit at the plug key. A deflection will be produced and the needle will take some time to come to rest. Students should therefore practise making connections in such a way that the needle is not set into violent 250 MAGNETISM AND ELECTRICITY LII oscillations, introducing for the purpose an extra plug key into the circuit close to the galvanometer. The following rules will be found useful : — If the needle is at rest in its normal position, send the current through the galvanometer for ^th of the com- plete time of oscillation of the needle, then break the circuit. The needle will continue to move but with diminishing velocity. When it is on the point of turning back, make contact again, this time permanently. If the needle moved without friction, it would be brought to absolute rest in its new position of equilibrium. During the first period the needle moves through half the ultimate angle of deflection, hence if the new position of equilibrium is approximately known, it will be easier to fix the end of the period by the position of the needle than by an estimate of time. Thus, if it is known for example, that the needle will be deflected through about 40°, make contact until the needle passes through 20°, then break circuit. The needle will continue to move owing to its inertia till it reaches 40°. As soon as it begins to turn back, make contact permanently. If the needle is deflected by a current which is to be interrupted, break the circuit until the needle is half way between its original position and the zero point, make the circuit again until the needle comes to rest at the zero, then interrupt permanently. Students are recommended to practise these rules till they are able to avoid producing large oscillations of the needle and consequent loss of time. The student should prepare his note-book for recording the observations by ruling columns according to the scheme given further on. The headings of the columns should be written out completely before the experiments are com- menced, so as to leave nothing but the actual numbers to be entered. Break the circuit, adjust the torsion head of the dynamometer till the pointer attached to the coil reads zero, read to 1° the position of the torsion head pointer, and of the ends of the galvanometer needle and record as shewn below. Re-make the circuit, adjust the torsion head till the pointer LI I COMPARISON OF CURRENT METERS. 261 reads zero, read to '1° the position of the torsion head pointer, then the position of each end of the needle of the galvano- meter, and finally again the dynamometer. Reverse the current and take the readings in the same order. In each set of observations there are, therefore, four readings to be recorded for each instrument. The galvanometer connections and resistances are to be varied as follows ; and observations taken as above : — I. Current passing through one turn of wire on the galvanometer (connect to terminals between which 1 T is marked). (1) 3 Resistance coils in series. (2) 1 Resistance coil only. (3) 3 Resistance coils in parallel. II. Current passing through 2 turns of wire on the galvanometer (connect to terminals between which 2 T is marked). The resistances to be varied as in I. III. Current passing through 3 turns of wire on the galvanometer (connect two previous sets in series). The resistances to be varied as in I. In order to compare the instruments, calculate the current indicated by the dynamometer from the formula A = Ksjd where d is the deflection and K is the constant of the instru- ment. Measure the outside diameter of the turns of ^vire of the galvanometer and the thickness of the wire. Calculate the mean diameter of the turns. Determine the value of the horizontal component of the earth's magnetic force at the point of observation, by taking the time of oscillation of a horizontal magnet at the point, and at one at which the value of the force is known. Calculate the current constant of the galvanometer. Determine the mean deflections of the galvanometer to two decimal places, and the currents by the formula il =7 tan to three figures, and tabulate as follows : — 252 MAGNETISM AND ELECTRICITY LII 29 April, 1896. Comparison of Tangent Galvanometer (A) with Siemens Dynamometer (A). Outside diameter of turns of galvanometer... Diameter of wire, including insulation Mean diameter of turn Mean radius of turn (r) Value of earth's horizontal force = 17 ( jTry) = {H) Calculated constant of galvanometer : 1 turn 2 turns .. * „ 3 turns = 19-86 cm. = -12 cm. = 19-74 cm. = 9-87 cm. •17. 2-67. 1-34. •89. Resist- ances 1 s Dynamometer. Constant=-15 Galvanometer 1 turn. Constant =2 -67 Ratio Read- ing Mean defltn v/def. Cd Readings Deflection 6 tand •89 tan e OdIC^ Wend Eend Wend Eend Mean P-5 zero 5n l°-Os zero 3 in series a ^24-2 i23-8 22-5 4-73 •711 15-5n lo^ls 15-0 14-1 14-55 •278 •722 •98 it /S \25-2 /24-8 23-5 4-84 •727 14-38 15-ln 14-8 15-1 14-95 -286 •740 -98 single iS i54-5 J54-7 »> a ^55-5 J55-3 Bin parall. a ^80 J80-3 >» ^ j78 177-6 1-5 zero Make out similar tables for the observation with two and three turns of wire on the galvanometer. SECTION LIIL VERIFICATION AND APPLICATION OF OHM'S LAW. Apparatus required : Tangent Galvanometer having a constant about 1, or an ammeter, voltmeter reading to 2 volts, 1 meter of No. 25 platinoid, No. 27 copper and No, 31 iron wire, 50 cms. of No. 29 platinoid vjire, all mounted on graduated boards, wire gauge. The law discovered by Ohm in 1827 states that the ratio of the difference of potential between two points of a conductor through which a current is flowing, to the current flowing, is a constant which depends only on the material, temperature, and shape of the conductor between the two points, and is indepen- dent of the strength of the current. It is the object of the present exercise to verify this law and to shew some of its applications. (if & ^ Pig. 103. J). ..-^n ^^m Fig. 103 a. Exercise 1. To verify that the difference of potential be- tween two points on a conductor is proportional to the current passing through the conductor. In order to verify this law, arrange (Figs. 103 and 103 a) a storage cell E in series with a four- way reversing key K with the plugs out, a tangent galvanometer or ammeter G, and a meter of No. 25 platinoid wire AB, stretched along a graduated 254 MAGNETISM AND ELECTRICITY LIII board. See that all connections are good, none of the wires being loose in the binding screws. Clean the movable knife- edges provided, place them on the platinoid wire at G and D 20 cms. apart, and connect them to the terminals of the volt- meter or high resistance galvanometer F, capable of reading up to 2 volts. Head the description of the instrument and the instructions for its use supplied with it. See that the pointer swings freely and stands at zero when the circuit through the instrument is broken. Make the circuit at K, read the deflection first of V, then of G, and again of F, estimating to tenths of a division. Reverse the current at K and repeat the observations. Insert a thin wire of copper into the battery circuit, in series with the platinoid wire, and repeat the observations of the deflections, keeping the contacts on the platinoid wire at the same points. Introduce further a thin wire of iron into the circuit and repeat the observations. Introduce lastly the short length of No. 29 platinoid and repeat the observations. Break the circuit at K. If necessary reduce the readings of the instruments to quantities proportional to currents, and record as follows : 12 September, 1897. Exp. Commu- tator Deflections Batios Voltmeter. Direct reading Tangent galvanometer Tangents 1 2 3 4 a b b a a b b a It] «•« •96G •933 •543 •368 26-7 26-8 26-2 26-5 The constancy of the ratio in the last column verifies the above statement. LIII VERIFICATION AND APPLICATION OF OHMS LAW 255 The constant ratio of the difference of potential between two points of a conductor, to the current which it produces in the conductor, is called the " Resistance " of the conductor between the two points. Exercise 2. To verify that the resistance of a homogeneous conductor of uniform cross section, is proportional to its length. With the cell circuit arranged as at the last observation, place one knife-edge on the platinoid wire at the division marked 10, and keep it there permanently during the experiment. Place the second knife-edge at the division marked 20, close the circuit at K, and keep it closed during the remainder of the experiment. Wait one or two minutes till the wires, which are being heated by the current, take up a stationary temperature, note the time, then take readings of both instruments. Move the knife-edge from 20, successively to 30, 40, 50, 60, 70, and 80, and then back again, reading the voltmeter at each position. Read the current, note the time, and record the observations as follows : 12 September, 1897. Deflections direct Time of first reading : 1 h. 43 m. rime of last reading: 2h. Om. Galvanometer 20*2 Galvanometer 20-1 Deflection of Knife Knife edge edge Length cms Voltmeter Mean Deflection Difference Deflection length A at Bat Ist series 2nd series 10 20 10 4-9 4-8 4-85 4-85 •485 30 20 9-8 9-8 9'8 4-95 -490 40 30 14-7 14-6 14-65 4-85 -488 50 40 19-8 195 19-65 500 •491 60 60 24-5 24-9 24-7 505 -494 70 60 29-5 29-6 29-55 4-65 •493 80 70 34-2 34-2 34-2 4-65 •489 ■ 1 If the strength of the current has altered during the experiment, the first and second set of readings will not quite agree. But if the change in the current has been uniform, and if the second set of readings has been taken in the reverse 256 MAGNETISM AND ELECTRICITY LIII order to the first, the mean of the two sets should give the readings con-esponding to the time half-way between the beginning and the end of the experiment. The column headed differences in the table of results, contains the differences between each pair of successive numbers in the previous column. These differences, which correspond to equal differences in the length of wire CD, should be nearly equal. Small deviations from equality are produced by : 1st, errors in reading ; 2nd, want of uniformity in the wire AB ; 3rd, errors in the voltmeter scale. If there is a large deviation it is probably due to one of the contacts being bad. Reverse the commutator in the cell circuit and repeat the whole series of observations, entering them in a second table headed — Deflections reversed. Exercise 3. To verify that the resistance of a homogeneous wire varies inversely as the area of its cross section. Without disturbing the circuit through the tangent galvano- meter, place the knife-edges on the No. 29 platinoid wire, at a distance of 10 cms. apart, and observe the deflection of the voltmeter. Now place the knife-edges 10 cms. apart on the No. 25 platinoid wire and again observe the deflection. Measure the diameter of each wire in the middle of the length between the voltmeter contacts, and enter the deflec- tions, the diameters and areas of the cross sections of the wires, as follows : Galvanometer reading , _i20'l. Wire Commu- tator Voltmeter deflection Diameter of wire Cross section Voltmeter deflection X section No. 25 No. 29 a h h a •0504 cm. •0352 cm. •00020 •000361 •0266 •0264 The last column shews that the above statement is correct. LIII VERIFICATION AND APPLICATION OF OHM'S LAW 257 Exercise 4. To find the relative specific resistances of copper, iron and platinoid. Keeping the main circuit undisturbed, place the knife- edges connected with the voltmeter on the copper wire 80 cms. apart, and take readings of both ammeter and voltmeter. Remove the knife-edges from the copper, place them 40 cms. apart on the iron wire, and read the voltmeter. Now place them 20 cms. apart on the No. 29 platinoid, and take readings. The ammeter is read in order to make sure that the current has not changed during the experiment and the following calculation is made on that supposition. Measure the diameters of the wires in the middle of the lengths used. From the observations calculate the deflections which would have been obtained with 1 cm. of each wire between the knife-edges. Then calculate what would have been the deflection if the cross sections of the wires of 1 cm. length had been 1 sq. cm. These deflections will be proportional to the specific resistances of the materials of the wires. Assuming the specific resistance of copper to be 1, calculate those of iron and of platinoid. Tabulate as follows : Jan. 5, 1900. Galvanometer Reading at beginning 20' 1 .. end 201 1 wire Length Voltmeter deflection Deflection per cm. Diameter Cro« section DeflecUon perrai. xeect. BeUUTe apeciflc resistance Copper 27 Iron 31 PUtinoid 29 80 cms. 40 „ b 2-6i 2^ •0326 •890 •988 •0417 •0296 •0862 •001.14 •000,682 •000.961 •000,0369 ■000,266 •000.898 1 72 248 8. P. 17 SECTION LIV. ARRANGEMENT OF CELLS. Apparatus required: Tangent galvanometer with a con- stant of '2 or '3, reversing key, four Daniell cells, resistance coils, and connecting wires. The Tangent Galvanometer may be used to illustrate the different ways in which voltaic cells may be joined together, and to shew which way is the most advantageous under given circumstances, but a more sensitive instrument than that used in the previous exercises will be necessary. Set up four Daniell cells, having porous pots as nearly alike as possible. If the porous pots are dry, allow them to stand fully immersed for half-an-hour in a bath of zinc sulphate solution, in order to fill the pores of the pots with the solution. Remove them from the bath, and place each in a jar containing a zinc plate dipping into zinc sulphate solution. Then fill each porous pot with concentrated copper sulphate solution, and place a copper plate inside it. Place the galvanometer with its coil in the magnetic meridian and read the zero. In order to find the resistance of each of the four Daniell cells, connect one cell through a commutator or reversing key. Fig. 104. LIV ARRANGEMENT OF CELLS 259 and a coil having a resistance of R ohms (about 4), with the galvanometer, and observe the deflection, (1) with the re- sistance of R ohms, (2) without the resistance, and (3) once more with the resistance in circuit, two observations being taken in each case, once with the current in one direction, and once with the current reversed. Read again the zero. Calculate the mean deflections and their tangents. If f/ is the mean of the tangents in (1) and (3), and ti is the tangent in (2), we have Kt,^ B-i-G-hR E where K represents the galvanometer constant, E the electromotive force, B the resistance of the cell, and G the resistance of the galvanometer, which for the present purpose may be neglected, as it is small compared to B and R. Putting G = 0, we deduce from the two equations Proceed in this way to determine the resistance of each of the four cells. Enter your observations and results as follows : — 23 Jan., 1894. Resistance of Daniell Cell 1. ReflisUnce in circuit ohms Readings Deflections Tangents of means Mean resistonce BMt West East West Mean 4 01 j 4-01 j •6ii 35-5 n 85-08 48-0 8 48-7 n 85-5 ri 86-08 •5n 85 2 8 85-0 n 48-3 n 48-5 8 85-08 84 -80 Zero 36 n 35-5 8 48-5 8 48-2 n 85 -On 85-58 Zero 35-2 8 35 On 48-3 n 48-5 8 85-08 84 -8 n 35-2 48-3 j 35-1 -705 l-122 = f, -703 •704 ^f/ * The mean defleotion can be obtained by taking the mean of the four readings of the ende of the pointer without making uee of the zero, but it is advisable in this exercise to adopt the method shewn in the example, as it enables mistakes to be more easily detected and corrected. 17—2 260 MAGNETISM AND ELECTRICITY LIV •704 .-. Resistance B^ of cell 1 = 401 ^ =675 ohms. Make out a similar table for each of the four cells. Connect all the cells in series through 8 ohms external resistance (Fig. 104), to the galvanometer, observe the de- flections and find the tangent of the mean angle of deflection produced by the current in the two directions. From the observed deflections with the separate cells, calculate the deflection which ought to have been obtained, as follows: — For the first cell we have — l~/{" and similar equations for the others. Hence i.e. if \ I I I ~ K - = -(- - 1 -^ t will be the tangent of the angle of deflection when the cells are all connected in series to the galvanometer. Hence if t' is the tangent of the angle of deflection with 8 ohms in series, / _ Bi-\- B^-\- B^-)r B^ ~ B^ + B^ + B, + B,-\-S' ' Calculated If = '649. Observed t' = '640. Arrange all the cells in parallel, and observe the deflections with 4 ohms in series with them. Calculate the tangent of the mean deflection. LIV ARRANGEMENT OF CELLS 261 Also calculate the deflection from the observations made with the separate cells, as follows: — For the first cell we have U E 1 ~ K b; and similarly for the other cells. Hence t^ + f^-hU-h u E K' 1.6. if t^ti + ti + ts + ti, t would be the tangent of the deflection if the cells were connected in parallel with the galvanometer. Hence if t' is the tangent required, for the case that the 4 ohms are inserted t t' = 1 1 «) + 1 \B, ' B^' B, Calculated ^'=1118. Observed ^'=1-120. Arrange the cells in two parallel sets of two cells in series, and determine the tangent of the deflection with 4 ohms in senes with the galvanometer. Establish a formula and calculate the deflection similarly. Determine by experiment which of the three arrangements of the cells previously tried, gives the greatest current through external resistances of 1, 4 and 8 ohms respectively. Tabulate the results as follows, underlining the maximum deflection in each External Mean deflections Cells in parallel In two sets in parallel In aeries 1 ohms. 65-3' 481 33-7 39-2' 30-2 21-6 42* 36-9 371 SECTION LV. THE WATER VOLTAMETER. Apparatus required : Water Voltameter, two storage cells, tangent galvanometer (with a constant of \ or 2), reversing key and resistance coils. When an electric current decomposes a liquid through which it is sent, the liquid is called an " electrolyte," and the process of decomposition is called " electrolysis." The laws of electro- lysis state that the weight of an element set free from a com- pound by electrolysis, is proportional to the current, the time, the atomic weight and inversely proportional to the valency of the element. One ampere flowing for one second liberates, e.g. •01118 grams of silver from any silver salt. The amount of an element liberated by a current in a given time may therefore be used to determine the magnitude of the current and hence to standardise a current-measuring instrument. In the present exercise a tangent galvanometer is to be standardised by means of a water voltameter. There are two ways of using a water voltameter, according as the hydrogen alone is measured, or both oxygen and hydrogen are collected together. In the latter case we gain by having a greater volume of gas to measure and being able therefore to reduce the time of the experiment, but there is some danger of error owing to the formation of ozone and its absorption in the water. It has however been shown by Kohlrausch that if currents of over one ampere are to be measured with an accuracy sufficient for commercial purposes, a voltameter, in which both gases are collected together, will answer the purpose. For smaller currents the hydrogen only should be measured. The difficulty may also be overcome by using a solution of sodium hydrate (between 5 and 10 "/o) with nickel electrodes. The volume of mixed gas produced by the passage of an ampere for a second is 1734 c.c. at normal temperature and LV THE WATER VOLTAMETER 263 pressure. The gas is collected in a graduated tube and its volume measured. To reduce the observed volume to the normal conditions, the temperature of the gas must be measured by a thermometer placed, if possible, inside the voltameter tube. The pressure is obtained by reading the barometer, by applying if necessary a correction for a, difference of pressure inside and outside the voltameter tube, and by deducting the pressure of aqueous vapour in the tube. As pure water offers a high resistance to the electric current a little sulphuric acid is added to it. If the reductions are to be made to the highest possible accuracy, account must be taken of the fact that the pressure of saturated vapour of water is less over a solution of sulphuric acid than over pure water. But in the present exercise it is assumed that the acid added to the water is not sufficient to produce a sensible effect. The voltameter provided (Fig. 105) consists of two portions, a lower reservoir and the graduated tube, which fits into it by Fig. 105. means of a giound joint. The lower reservoir communicates with the atmosphere through an opening which can be closed by a glass stopper. During electrolysis this stopper must be removed. Pour water acidulated with about 15% of dilute sulphuric acid into the reservoir so as to render it about three- quarters full. Insert the tube and stopper and tilt the volta- meter so that the liquid runs into and completely fills the tube. Place the instrument in a tray and remove the stopper. Connect the galvanometer and voltameter in series with a reversing key, a resistance and two storage cells. The object of 264 MAGNETISM AND ELECTRICITY hV the reversing key is to correct for errors of zero of the tangent galvanometer, by taking its deflections during half the time in one direction and half the time in the other. The key might therefore be inserted so as to affect the galvanometer only, leav- ing the current through the voltameter in the same direction throughout the experiment. Adjust the resistance so that when the circuit is closed, the deflection of the galvanometer needle is about 45". When this has been done, interrupt the current at the key, and wait till the bubbles of gas in the upper part of the column of liquid have ascended into the space above. Then read the position of the top of the column on the tube, measure to the nearest cm. its height above the liquid in the base, and note the temperature of the gas. Read the zero of the galvanometer. Now make the circuit at the key, noting the time, and at the end of a minute read the galvanometer, repeating the reading every two minutes till the tube is about J full of gas. One minute after taking the last reading, break the circuit, noting the time. After allowing the bubbles of gas in the column of liquid to ascend, read the position of the top of the column and measure as before the height of the liquid column, and the temperature of the gas. At a given instant make the circuit again, so that the current passes in the opposite direction through the galvanometer for an equal time. Read the galvanometer every two minutes as before, and the final position of the level of the liquid. Read the barometer. Calculate for each of the three observations of the column of liquid, the volume of gas reduced to normal temperature and pressure, and subtract the first from the second and the second from the third. The diflerences are the volumes of gas produced in the observed times by the passage of the current. Take the mean of the tangents of the galvanometer deflec- tions for each period. Then if d be the mean tangent, and K the constant of the galvanometer, the average current passing is Kd amperes. If v c.c. of gas were liberated in t seconds, then v=V. Kd . t, where V is the volume of mixed gas liberated by 1 ampere in 1 sec, both being measured at the same tempera- ture and pressure. LV THE WATER VOLTAMETER 265 If V and V are the volumes reduced to a pressure of 76 cm. and a temperature of CC, we may substitute for Fits numerical value 1734 and thus find V K = Calculate K from the observation in this way, arranging the work as shewn below. The two sets of observations are reduced separately so as to afford a check on the calculations and a test of the consistency of the results. Remove the tangent galvanometer, and substitute for it one of the boxes used in the Magnetic Survey of the Laboratory (Section L.). Determine the time of oscillation of the needle. Determine also the time of oscillation of the needle when placed at some point of the Laboratory at which the earth's horizontal magnetic force H is known, and from the observations calculate the value of H at the place where the galvanometer stood. From the value of H and the radius and number of turns of the coil calculate the constant of the galvanometer (see Section LL), and compare the calculated value with the result of the experiment. Arrange your observations as follows : — 17 March, 1898. Voltameter A. Tangent Galvanometer B. Height of Barometer = 75*67 cms. Voltameter observations I n 133-5 m 1 Heading of voltameter tube*, c.c. 16-5 253-8 Heiglit of column of water, cms. 35 26 15 „ „ equivalent mercury column 2-58 1-91 111 Pressure of ga.s, cms. 7309 73-76 74-56 Temperature (Centigrade) 18° -8 18''-7 18''-5 Corresponding va|K)ur presHUi*e, cms. 1-61 1-60 1-68 Presssure of dry gas, cms 71-48 7216 72-98 Volume of gas reduced to 76 cms. and O'C, ac 14-5 118-6 228-7 * If the tube is not divided into oubio oentimetret, the value of the divinont onght to be ascertained by experiment. 266 MAGNETISM AND ELECTRICITY Galvanometer Observations I. LV The readings of the ends of the pointer, marked A and B, are taken as positive when the deflections are counter-clockwise as measured from the zero divisions of the scale. Time Readings of Pointer Deflections Tangents End A End B Mean 11 h. 57 m. + o°-o + 0°-2 H-O^-l 11 58 12 00 02 04 06 08 + 46-6 46-2 45-5 45-5 45-3 44-9 + 46-6 46-3 45-7 45-7 45-5 451 + 46-6 46-2 45-6 45-6 45-4 45-0 46-5 46-1 45-5 45-5 45-3 44-9 1-054 1-041 1-018 1-018 1-010 0-996 09 + 0-0 + 0-2 + 0-1 Mean = 1-023 Current started at 11 h. 57 m. „ stopped at 11 h. 08 m. Similarly for the observations during the second half of the experiment. The results are tabulated as follows : — Volume generated Time Mean tangent K 118-6- 14-5 = 104-1 228-7-118-6 = 110-1 660 sees. 650 „ 1023 1-050 -89 -93 Mean = •91 Constant of Galvanometer by calculation Time of oscillation at place of observation ... = 7*12 sees. „ „ „ standard position .. . ... = 7-16 „ Value of jy „ „ „ = -171 „ Value of ^ at place of observation ... ... = '173 „ Radius of Coil of Galvanometer ... ... =9*7 cms. Number of turns ... ... ... ... ... = 3 .'. Constant = ... ... ... ... = "89 SECTION LVI. STANDARDISING CURRENT METERS BY THE COPPER VOLTAMETER. Apparatus required : Current-measuring instrument to he standardised (an instrument intended to measure accurately one or two amperes is suitable) ; storage cells, copper depositing cell, copper electrodes, plug-key, resistance coils, accurate balance. In the previous exercise the amount of electrolysis produced by the passage of a current was estimated by the volume of gas generated. More accurate results are obtained by the use of an electrolyte of which one of the products of decomposition is a solid which can be weighed. Silver and copper salts are found to be suitable for the purpose, as secondary reactions may be more completely avoided than in the case of other metals. For the most accurate work the silver voltameter is used*, but its manipulation requiring great care, and the materials being ex- pensive, copper may be substituted, and with proper precautions an accuracy of one part in a thousand may be attained. In the present exercise a solution of copper sulphate in water will be used, and a current meter will be standardised by the weight of copper deposited. The example is worked out on the supposition that the instrument to be standardised ifl a Kelvin Current Meter, but any other accurate instrument such as that shewn in Fig. 106 may be substituted. Prepare a 20% solution of copper sulphate, by adding 120 grams of copper sulphate crystals to 480 grams of water. When the crystals have dissolved, filter into the beaker provided and add one or two c.c. of strong sulphuric acid. • Lord Rayleigh and Mm Sidgwick ♦•On the electrochemical equivalent of silver and on the absolute electromotive force of Clark cells," Phil. Tram. 1884. 268 MAGNETISM AND ELECTRICITY LVI Clean the three copper plates provided, two of which are to serve as anodes and the third as cathode, with sandpaper. Dip the one which is to serve as cathode into dilute nitric acid for about three minutes, then remove it and place the three in dilute sulphuric acid for about three minutes, finally wash under a stream of tap-water. Place the first two plates in the side clips A of the stand (Fig. 106) provided, and the third plate in the centre clip. Fig. 106. Join up the copper voltameter, a plug-key K and an adjust- able resistance to the instrument to be standardised, through two storage cells (one of which only is shewn in the figure) or other battery of small resistance giving about 4 volts. Be careful to connect the terminals of the battery in such a way, that the current passes from the outside plates (anodes) to the central plate (cathode) of the voltameter. If the instrument to be standardised gives deflections in different directions when the current is reversed, a reversing key should be inserted so that the directions of the current can be changed rapidly in the current meter without being reversed in the voltameter ; and that change should be made in the middle of the experiment. Adjust the resistances until the current is of convenient amount to be measured. Raise the plates out of the solution, take the copper cathode from its clip, wash it in clean water, and dry first in a sheet of filter-paper, then before a fire, heating coil, or gas flame, taking care not to heat the plate appreciably. When the plate is dry and at the temperature of the room weigh it carefully. LVI COPPER VOLTAMETER 269 As the absolute weight is required, the plate must be weighed on both sides of the balance according to the method describetl on p. 45, and trustworthy weights only must be used. Read the zero of the instrument. Replace the cathode and lower the copper plates into the solution, make the circuit at a time to be noted in minutes and seconds, and as soon as possible take a reading of the instrument to be standardised. Take readings every five minutes for forty-five minutes or an hour. The true readings will be the differences between the actual readings and the mean of the zeros read at the beginning and end of the experiment. Note the temperature of the electrolyte. While the electrolysis is going on, clean a beaker, fill it with clean water, and add a few drops of sulphuric acid. At the end of the above period break the circuit, noting the time accumtely, and read the zero of the instrument. Raise the electrodes, remove the cathode and dip it as quickly as possible into the acidulated water, then hold it under a gentle stream of water from the tap for a minute. Now dry it as before, first in a pad of clean filter-paper, then before a fire, heating coil or flame, and after cooling weigh again accurately. The relation between the current (7, the electrochemical equivalent of copper z^ and the weight W deposited in a given time t is W W--C.z.t, .-.(7= ,. z . t In the absence of all secondary reactions the value of z should be constant. But owing probably to the presence of dissolved oxygen, the amount of copper deposited is not strictly proportional to the current and depends to a small extent on the temperature. These effecta can be taken into account by making z depend on the current density at the cathode and on the temperature. Thomas Gray, who has carefully investigated the amounts of copper deposited under different conditions, has 270 MAGNETISM AND ELECTRICITY LVI given the following values for the electrochemical equivalent of copper. Area of cathode in square centimetres per ampJsre of Temperature 12° Temperature 23° current 50 •0003288 •0003286 100 •0003285 •0003283 150 •0003282 •0003280 200 •0003279 •0003277 The results of this Table may be expressed by the equation A KQ f TO •0003288 - 0000003 ,^ - 0000002 --— , 50 11 where A is the area of the cathode surface per ampere, and t the temperature. The value of A is found by calculating the current in the first instance approximately, using ^-^ as the equivalent. Measure the total area of the two sides of the cathode, and obtain an approximate value of the area per ampere. From this and the temperature during the experiment find the equi- valent which applies to your experiment. B,ecord and reduce the observations as follows : — 15 June, 1899. Weight of cathode before electrolysis .. = 20-3798 grams „ ,, ,, atter ,, .. = 21-5786 „ Amount of copper deposited . = 1-1988 „ Duration of experiment =lh. Om. 25 s. . = 3625 seconds ^ ^ . ^ , 1-2 X 3000 Current approximately -q^qq- . = 1 ampere Total area of cathode = 66 sq. cm. Area of cathode per ampere of current . . = 66 Temperature . = 18° C. Equivalent of copper .. = •0003286 ^ 1-1988 Current =.^^32gg^ 3^25 - ' • = r0064 ampere LVI COPPER VOLTAMETER 271 Time Beading of Kelvin balance Zero True reading Square root llh. Om. current made 5 10 15 50 55 12h. Om. 25 8. current broken 409-6 406-6 406-4 406-4 405-4 405-6 406-4 •2 •2 409-4 406-4 406-2 405-2 405-2 406-4 406-2 20-23 20-16 20-15 20-13 20-13 20-16 20-15 MeaD = 2014 Constant of balance = Mean current 10064 Mean square root of deflection 20*14 = 04997 Constant given by instrument maker ... ... = 05000 Thus currents measured by the instrument using the con- stant -05 would agree to within one part in a thousand with their value as determined by copper electrolysis. The constant supplied by the maker of the Kelvin balance is equal to the mean current divided by twice the square root of the reading ; i.e. '020 in the case of the above instniment. A table of doubled squared roots is provided with the instrument, and the last column in the above Table may be replaced by one in which the double square roots are entered. But the method here adopted is applicable to all instruments of the dynamometer type and the square roots are easily found in Barlow's Tables*. If the instrument to be tested is a direct reading one, the last column is unnecessary, and if it is of the tangent galvanometer type it must be replaced by the tangents of the angles of deflection. * If the square root of e.g. 408-4 is taken from Barlow's tables, interpolation would seem necessaiy between the values of the roots given for 408 and 409, but this may be avoided by finding that number in the neighbourhood of 2000, the square of which has for its first four significant figures 4084. The ponition of the decimal point is obvious. SECTION LVII. ON MIRROR GALVANOMETERS AND THEIR ADJUSTMENT. In Section LI. it has been explained that the sensitiveness of a tangent galvanometer may be increased by diminishing the radius of the coil through which the current is passing; but that if the tangent law is to hold good the length of the needle must also be diminished. In many experiments it is of greater importance to have the galvanometer as sensitive as possible, than that the tangent law should hold accurately. We use in such cases a galvanometer, the coils of which have as small a radius as possible, leaving only just sufficient room for the magnet to move inside. The movement of this magnet is rendered apparent, and measured if necessary, by the help of a mirror attached to it, in the way explained in Section XXXIY. Several small magnets are often used instead of a single oue in order to increase the magnetic moment, and these are either attached directly to the back of the mirror by means of a little shellac, or the mirror is placed outside the coil and attached to a thin wire which passes through the coils and carries the magnets at its end. The suspended system must be free to move round a vertical axis, and is for this purpose attached to a fibre just strong enough to carry its weight and as free from torsion as possible. The deflection of a galvanometer for a given current being increased by a diminution of the intensity of the magnetic field at the centre of the coil we have a further means at our disposal of rendering a galvanometer more sensitive, or of ad- justing its sensitiveness to any desired value within certain limits. For this purpose a permanent magnet is used which together LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 273 with the earth's field produces a resultant field of the desired strength. A similar magnet is also necessary, when the galva- nometer has to be set up in such a position that the plane of the coil does not lie in the magnetic meridian. The strength and direction of the earth's field being given, it is theoretically always possible to produce a resultant field of any desired strength and direction by placing a permanent magnet with a sufficiently great magnetic moment in a proper position. When gi-eat sensitiveness is required, and at the same time the direction of the field is to be altered, it is generally advisable to proceed by two steps. With a magnet of known moment, it will not be difficult to find by calculation a position for it such that it will nearly neutralise the earth s field. A second weaker magnet may then be used to regulate the strength and direction of the field. A galvanometer made sensitive by counterbalancing in this way the greater part of the earth's force by means of external magnets has the disadvantage of being very sensitive to slight magnetic disturbances either caused by actual changes of the terrestrial forces, or to disturbing currents (electric trams) or to the accidental dis- placements of magnetic material (keys, spectacles, corset steels) which are almost unavoidable in a laboratory. To get rid of that portion of these effects which is nearly uniform in the space occupied by the galvanometer needles, so called astatic magnetic systems are often used. These consist of two mag- nets or two sets of magnets, which have magnetic moments of nearly equal value, and are rigidly connected with each other, so that the similar poles of the two sets point in opposite directions. One of the sets is placed in the centre of the galvanometer coil, the other above or below it, and in some instruments the second set will be placed in the centre of a second galvanometer coil in all respects similar to the first. The whole of the combination of magnets, called an astatic system, will behave like a very weak magnet towards the earth's force, and may set in a direction which it is impossible to predict. To make this latter point clear, let H represent the earth's field, m^.m^ the magnetic moments and a,, Og the angles between 8. p. 18 274 MAGNETISM AND ELECTRICITY LVII the magnetic meridian and the magnetic axes of the two sets. The mechanical couples exerted by the earth's field will be Hm^ sin a-y and Hnu sin a.^ and in the position of equilibrium mi sin «! + W2 sin ao = 0. If the angle between the two magnetic axes is very nearly equal to two right angles we may write ^2 = «! + TT + 5 where 8 is a small angle, so that sin as = — (sin oti -t- 8 cos a^). Hence for equilibrium lai h 7?li - ma nil = mo «! IT ~ 2 ' If it follows that hence the astatic system will set at right angles to the magnetic meridian) if on the other hand S = but m^ differs from mj, oti = and the system will set in the meridian. As in practice neither h nor m, - m^ vanish, but will be small, «] depends on the ratio between two unknown small quantities. Directing magnets will have to be used with an astatic system, in order to bring it into its proper position with respect to the galvano- meter coils and to regulate the strength of the field. The increased sensitiveness of an astatic galvanometer de- pends on. the fact that while the two sets of magnets oppose each other in so far as the earth's directing force is concerned, the electric current acts on both in the same direction. If there are two galvanometer coils, the current is led through them in opposite directions, so that the couples exerted by the currents in both coils have the same direction. A galvano- meter with two coils can also be used as a "differential galvano- meter," when it is required to test the equality of two currents or to measure very small differences between them. The two LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 275 cun-ents are in that case sent separately through the two coils, so that their eflfects on the suspended system of magnets oppose each other. While the observer has it in his power to increase consider- ably the sensitiveness of a galvanometer by weakening the magnetic field, he must remember that in so doing, he will increase the peiiodic time, i.e. the time of oscillation of the needle, and thereby render its motion more sluggish. This to a great extent counteracts the advantage gained by the increased sensitiveness, especially when owing to disturbing causes the zero of the needle is a little unsteady, for it becomes in that case impossible to distinguish a true deflection from an accidental shift. Therefore in modem galvanometers intended to be highly sensitive, great importance is attached to having the suspended magnetic system as light as possible, a small moment of inertia allowing a corresponding weakening of the directing field without increase of the time of oscillation. In the case of a D'Arsonval galvanometer or any instrument with a suspended coil, it is not generally in the power of the observer to alter the sensitiveness. Students should pay careful attention to the optical arrange- ment on which the measurement of the angles depends. If the objective method is used, they should secure a good image of a wire on the scale, and in every way try to increase the ac- curacy of reading. It must be remembered that doubling the accuracy of reading doubles the effective sensitiveness of the galvanometer just as much as halving the strength of the directing field, without introducing the disadvantage of increased peri<xlic time. Insufficient care is often given by instrument makers to the mirror of a galvanometer, which is a more im- portant part of the instniment than many others, to which they devote great attention. The image formed should be as perfect as the size of the miiTor will allow; the latter should be either plane or concave with a radius of curvature of about one metre. If the instrument is home made, microscope cover glasses are generally silvered for the purpose, and among a number of them, some are always found which give sufficiently good images before mounting, but after the magnets have been 18—2 276 MAGNETISM AND ELECTRICITY LVII fixed to the back of the mirror, it often turns out to be useless on account of the distortion produced by the material which is used to attach the magnets. It is therefore better to fix these in the first place to thin mica, and to attach the mirror to the other side of the mica by means of a small drop of shellac holding the mirror, at one point only. If the mirror hangs out- side the galvanometer coil the difficulty does not arise. The curvature of the mirror, and the definition of the image formed by it, may be investigated by using a source of light some distance away, and determining whether there is a good re- flected image within a few feet of the mirror. If there is, the distances of object and image will determine the radius of curvature. If the mirror is plane, or nearly so, the image may not be real or may be too far away, and in that case, must be looked at through a telescope. The telescope being focussed for the image, we may easily determine the distance of a point which when looked at directly is also in focus. From this and the distance of the source of light, the focal length of the mirror may be determined. The principal adjustments of mirror galvanometers consist according to the above explanations, (1) in securing that the suspended system can turn freely round a vertical axis through the largest angles on either side of the zero position, which are likely to be used during the observations, (2) in securing that the resultant force should be nearly parallel to the plane of the galvanometer coils, and (3) in adjusting the strength of the field to the required intensity. If it is desired to adjust a galvanometer ah initio, the first step should be to place the galvanometer on a firm support, if possible in the centre of the room so that access may be had to it from all sides. Freedom of motion of the galvanometer needle should be obtained by adjusting the level of the in- strument. If it cannot be secured thus, carefully notice what is the cause of the impediment. Possibly the needle hangs too low or too high, and in that case the suspension must be altered, but this should only be done by experienced hands. When the needle is free, observe its approximate position of rest. With non-astatic systems it should be in the direction LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 277 of the resultant field at the place of observation. If this is not the case, the fibre is probably twisted. Unless the angle of twist is very great, no serious error will result ; but instruments should be provided with some means of turning the suspension head, and thus untwisting the fibre, so that the needle may hang in the proper direction. It is always an advantage to be acquainted with the peculiarities of each instrument, and the effects of torsion should be noted by twisting the upper end of the suspension through a measured angle and noting the angle through which the suspended magnet turns in conse- quence. This angle of twist will prove useful in the future use of the instrument, because if the suspended system is losing its magnetisation, the angle will gmdually increase, and shew when the time has arrived for taking out the suspension and re magnetising the magnets. The time of oscillation T of the magnets when swinging in the earth's field should also be noted. If M is the magnetic moment, H the strength of the field, / the moment of inertia and T the torsion per unit angle of twist we have the relation y HM- and if a twist through an angle a at the upper end of the suspen- sion has produced an angular displacement 6 of the magnet system : T{a-e) = HM^md, where o will in general be sufficiently great compared to 6 to allow the latter to be neglected on the left-hand side of the equation. If the suspension has been put together in the labo- ratory, the weights and dimensions of different portions of it should have been noted, so that the moment of inertia can be calculated with sufficient accuracy. In that case, the above relations will allow the determination of J/ and t. The value of t is of some interest in instruments intended for delicate work, for it determines the limit beyond which it is not possible to push the sensitiveness of the galvanometer, even if the earth's field be wholly neutralised. 278 MAGNETISM AND ELECTRICITY LVII The time of oscillation and the direction of the needle having been noted, the position where it is desired to set up the galva- nometer must be considered. This will generally be along a wall of the laboratory, or at right angles to it, and therefore not necessarily in the direction of the magnetic meridian. A rough calculation will give the strength of the required magnetic field which together with that of the earth will set the magnet parallel to the galvanometer coils, and give a time of oscillation which should not be greater than 8 or 10 seconds. If the focal length of the mirror is about a metre, the scale may be set up at that distance from the mirror, but if the mirror is plane a convex lens of one metre focus should be fixed to the galvano- meter as near to it as possible. If the mirror is neither plane nor has the required radius of curvature, the focal length of a lens must be calculated so that, when fixed near the mirror, a point of light at a distance of one metre should give a real image at the same distance. The scale is now set at the proper distance, and the last adjustment of focus is made either by altering slightly this distance or by introducing a weak lens to alter the divergence of the beam incident on the mirror. A thin wire placed in the incident beam, which may also be made movable for greater facility of focussing, serves as the object, the image of which on the scale determines the angular position of the mirror. Very often the preceding adjustments have been made for the student, at any rate approximately, and in that case his first care should be to check the adjustment for focus and to improve it when necessary. If the instrument is to be used not simply as an indicator but to measure deflections, the position of the scale must also be looked to. It should be placed at right angles to the line joining its centre to that of the mirror. This is most easily done, as explained in Section XXXIV. by measur- ing the distance from the ends of the scale to the mirror, or if the galvanometer is covered by a glass shade, it will be sufficient to measure the distances from that point of the shade, which is estimated to lie on the central line. The investigation on page 159 shews that if an accuracy of one per cent, in the readings is aimed at, and the means of deflections to right and left are LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 279 always taken, the scale may be inclined as much as 8 Vo without doing any harm. The difference in the distances of the ends of the scales from the mirror should in that case not exceed the seventh part of the length of the scale. If however the deflec- tions to either side should be correct to within one per cent, a ten times greater accuracy is required in the adjustment of the scale and this requires some care in the measurements. The last adjustments relate to the sensitiveness of the instrument. If this is insufficient a permanent magnet should be placed parallel to the galvanometer needle and so that the line joining the centres of the two magnets is approximately perpendicular to their magnetic axes. If the poles of the out- side magnet point in the same direction as those of the sus- pended needle, the magnetic field will be weakened by it. The magnet is first placed at a distance and then brought slowly nearer, the spot of light being watched at the same time, so as to keep it near the centre of the scale. It will be found that when the outside magnet is brought too near, the position of the galvanometer needle becomes unstable and tends to be reversed. When the magnet is placed so far away that the needle is just stable, the instrument is as sensitive as it can be made, but it will probably not be advantageous to use it in this most sensitive condition owing to the increase in the length of the time of oscillation. In some instruments the controlling magnet is clamped to a vertical rod fixed to the galvanometer stand, and its position is altered by loosening the clamp and sliding the magnet up- wards or downwards. Some makei-s have adopted a method in which two controlling magnets at a fixed distance are used. The field due to the two magnets may be strengthened or weakened by altering the angle between their magnetic axes. In the case of galvanometers having astatic systems, the sensi- tiveness may be altered in the same manner by an outside magnet. Delicate galvanometers may be damaged when too strong a current is sent through them ; and very often it is not possible at the beginning of an experiment to make sure that the elec- tromotive force at the terminals of the galvanometer does not 280 MAGNETISM AND ELECTRICITY LVII exceed the limit of safety. In that case a "shunt" should be used. This is a resistance smaller than that of the galvanometer and connected to it in such a way that the current will pass through the galvanometer and shunt in parallel. Sometimes a number of shunts are supplied with the instrument having their resistances graduated so that only the j'^, yj^ or jj^jj^th. part of the current may pass through the galvanometer. The sensitiveness of a galvanometer is measured by its power to indicate small variations of current, but it does not follow that the most sensitive galvanometer is the one that indicates the smallest variations of the quantity to be measured. On the contrary a galvanometer, which in the ordinary meaning of the word would not be called " sensitive " at all, is sometimes the one which will give the most accurate results. As it is very important that the student should clearly realise this, it is necessary to enter a little further into the theory of the galvano- meter. Supposing the annular space, which is to contain the galvanometer coil is given, we may fill it either with thin or with thick wire ; in the former case we shall get a " sensitive " galvanometer, but one having high resistance, while the thick wire will give a low resistance but smaller sensitiveness. It is on the relation between the resistance and the sensitiveness that the theory of the galvanometer depends. If starting with a thick wire, we wish to replace it by one of half the diameter, we should be able to replace each turn by four of the thinner wire. But four turns of wire having one quarter the cross section will have sixteen times the original resistance. Hence while we have increased the sensitiveness four times we have increased the resistance sixteen times. In a similar way it may be proved quite generally that — neglecting the space lost by the insulation of the wire — the sensitiveness of galvanometers having the identical spaces filled by windings, will vary as the square root of the resistance. Writing k^/G for the sensitiveness of a galvanometer where G is its resistance and k a constant depending on the shape and size of the annular space containing the windings, let a circuit containing an electromotive force E be formed by the galvanometer and a resistance R, the current will be JE/(Ii + G) LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 281 aud the deflection of the galvanometer fcE*JG/{R + G). The deflection of the galvanometer will therefore be very small when the resistance of the galvanometer is very small and also when it is very large, and it may easily be shewn that the deflection will be a maximum, when G = R. Hence when small electromotive forces have to be measured, that galvanometer will be most delicate which has the same resistance as the external resistance. For thermoelectric work with circuits which generally have very small resistances, so called "sensitive" galvanometers will not be sensitive at all, and low resistance galvanometers will give the best results. When the cii'cuits are more complicated as in the Wheatstone bridge arrangement the above investigations have to be extended, if it is desired to find the most suitable galvanometer, but the general rule will be found to hold, that when the external resistances are small, the galvanometer resistance should be small also. In the case of galvanometers, in which the poles of the sus- pended magnets come close up to the windings, it cannot be assumed that the currents are proportional to the angles of de- flections, even when these are small. If therefore small currents passing through the instrument are to be compared with each other, the instrument should be calibrated. This is most readily done by the method used in Section LIII. (Exercise I.), a current being sent through a uniform wire stretched along a graduated scale and a suitable resistance. If the terminals of the galva- nometer are brought into contact with the wire, the resistance of which should be small compared to that of the galvanometer, the currents passing through that instrument will be propor- tional to the length of wire included between the terminals. A short description may be given in conclusion of the "damping" of galvanometers. A current passing through the instrument will deflect the needle, which will begin to oscillate about its new position of equilibrium. The oscillations follow the same laws as those of a pendulum, and will gmdually di- minish owing to frictional resistance and electrical damping. In many cases it is desirable that the needle should come to rest quickly, for which purpose it is advisable to increase the resistance, either mechanically by attaching a vane to the 282 MAGNETISM AND ELECTRICITY LVII suspended system, or electrically by bringing a mass of well- conducting material (copper) near the oscillating magnets. The currents induced in the copper by the motion of the magnets react on the latter and oppose its motion (Lenz' law). When the damping is so great that the needle will not oscillate at all, but gradually takes up its new position of equili- brium, without passing beyond it, the motion of the needle is called "aperiodic," and the galvanometer is said to be "dead beat." When the instrument is used to measure the quantity of electricity conveyed by a large current of short duration (dis- charge of a condenser, induction kicks), we require the deflection which would be produced if there were no friction, and in that case the damping should be small, and must be taken into account (see Section LXIL). A galvanometer used in this fashion is called a ballistic galvanometer. The angular dis- placement of a dead beat galvanometer is represented b}^ the expression c(l-e-^% where c is the ultimate displacement, which as is seen, will be reached theoretically only after an infinite time. The angular displacement of a damped needle, measured from its position of equilibrium, is represented by the equation 00 = ae--^^''^ cos K{t — U), T being the time of oscillation, and t^ any time at which the needle has its greatest positive elongation. In case the numerical value of \ has to be determined, suc- cessive turning points have to be observed. If tx is the time of the first observed positive elongation {x^ the successive positive elongations will take place at times ti + T, ti-^ 2T etc. and will therefore be given by a?3 = ae-'-'^(^+2r,/r LVII MIRROR GALVANOMETERS AND THEIR ADJUSTMENT 283 Taking logarithms on both sides and then diiferences between successive equations we obtain log Xi — log x» = 2\ log e log Xi - log a^s = 2\ log e log .Tj — log x^ = 2\ log e. Hence the differences between the logarithms of successive elongations are constant and the quantity X is called the " loga- rithmic decrement." The value of X determined in this way may be affected by a considerable error if the zero reading of the galvanometer changes progressively. This eiTor may be eliminated by measuring the elongations on both sides and taking the mean of the values of \ as found from the positive and negative elongations. A manner of reducing the obser- vation, which is better, because it saves some arithmetical labour, consists in measuring successive arcs of swing from right to left, left to right and so on. Calling these successive arcs: s^, 5«, etc., we find log s, - log Sn = (n-l)X log e log 52 - log Sn+i = (« - 1) \ log e, so that a series x>f independent values of X may be obtained the mean of which is taken for the final result. When the oscilla- s ^ s tions diminish slowly so that the square of ^— " may be neglected, the first of the above equations simplifies to ~(n-l)Sn' The quantity \ occurs chiefly in physical measurements, when the first deflection of a galvanometer needle is observed and it is desired to calculate what that deflection would have been in the absence of all damping. As the time taken by the needle to pass from the position of rest to its first elongation is T/4(, the above equations give a = Xie~^^^, where x^ is the observed elongation and a the required amplitude. If X be small this reduces to an equation which we shall have occasion to use. 284 MAGNETISM AND ELECTRICITY LVII Exercise I. Set up the mirror galvanometer in the given position and measure the time of oscillation of the suspended magnet, when oscillating under the action of the earth's force only. Double the time of oscillation by means of an external magnet. Determine the electromotive force necessary to pro- duce an angular deflection of 'Ol. Hence, assuming the tangent law to hold, calculate, using the given value of the galvano- meter resistance, the magnetic field produced at the poles of the suspended system per unit current passing through the galvanometer coil. Exercise II. Calibrate the galvanometer indications, and prepare a table shewing the relation between the deflection of the mirror and the current passing through the instrument. SECTION LVIIL MEASUREMENT OF RESISTANCE. . THE POST OFFICE FORM OF WHEATSTONE'S BRIDGE. Apparatus required : Galvanometer and scale, Post Office resistance boa-, Leclanche cell, coils, voltmeters, platinoid mire, and connecting luires. The students are supposed to be familiar with the principle of a Wheatstoue Bridge, and to have had some practice in the measurement of resistance by the simpler forms of bridge arrangement. When two of the resistances of the bridge are formed by a stretched wire, with a movable sliding contact for the junction leading to the battery or galvanometer, incon- veniences arise owing, (1) to the fact that the sum of the two resistances is fixed, being the total resistance of the wire, while increased accuracy could often be obtained, if that resistance could be varied according to the value of the resistance to be measured ; (2) to the uncertainty of the measurements, due to the possibility of different parts of the wire having unequal resistances, or the contacts being faulty; (3) to the arith- metical labour involved in reducing the measurements, which is of no importance if the instrument is only used occasionally, but becomes irksome by accumulation if many measurements are taken ; (4) the want of portability of the instrument. The requirements of telegraph offices, which have been the cause of many improvements in electrical instruments, have led to the construction of a form of a bridge consisting of a series of 286 ELECTRICITY AND MAGNETISxM LVIII resistances, appropriately connected, and compactly placed to- gether in a box which is known as the " Post Office box " (Fig. 107 a and b). Fig. 107 (I. Fig. 107 b. To understand its construction students are referred to the diao^ram of the bridge (Fig. 108) in which the battery and galvanometer circuits are brought to spring contacts at K^, Kc,. These spring contacts are placed in the box as shewn in the figures. There is a metallic connexion inside the box, not shewn in the figure, between C and Ko and also between A and K^. The letters in Figs. 107 6 and 108 correspond, so that if the resistance to be measured is placed between B and D, and the battery and Fig. 108. galvanometer connexions are made as shewn in the figure, the bridge arrangement is complete. The arms A G and BC include in general, resistances of 1000, 100 and 10 ohms respectively in <3ach branch, so that P and Q (Fig. 108) may either be equal in three different ways ; or in the ratio of 10 : 1 in two different ways ; or in the ratio of 100 : 1 by making one resistance equal to 1000 and the other equal to 10 ohms. According to the resistances placed in the arms P and Q, a resistance should be placed in the arm R, which, if P is equal to or greater than Q, should be >S', 10>Sf or 100>S, and if P is smaller than Q, 'IS or 'Ol^Si. Another variation can be made by inter- LVIII MEASUREMENT OF RESISTANCE 287 changing the battery and galvanometer, and it is therefore seen that a great many combinations for producing balance in the bridge are available. But these combinations are not equally good, the unknown resistance being capable of mea- surement with greater accuracy for a pai'ticular choice of P and Q, depending on the value of the resistance to be measured, and on the resistances of the galvanometer and battery circuits. In nearly all cases in which the highest obtainable accuracy is not required the following rules, if attended to, will prove a sufficient guide to the student in the selection of a proper combination of resistances ^ (1) As the resistance R between A and i) in the ordinary P.O. box can be varied in steps of one ohm, and as one-tenth of that amount may with certainty be estimated by interpolation of galvanometer deflections, resistances of over 100 ohms can be measured with an accuracy of at least '1.7o, if the arms P and Q are made equal to each other. Hence if a rough measure- ment has shewn that 8 has a value of more than 100 ohms, make P and Q equal to each other. If S lies between 100 ohms and 10 ohms and has to be measured to less than l^o. P should be 10 times as great as Q, while if S has a smaller value than 10, P should be 1000 ohms and Q 10 ohms. (2) When P is equal to, or ten times as great as Q, the best resistance of P depends on the galvanometer and battery resistances. For the exercises in this Section, any value will give sufficiently good results. It is easier in each particular case to vary P and Q and find the most favourable combination by experiment, than to determine it by calculation. (3) If the galvanometer has a greater resistance than the battery, connect the galvanometer circuit, so as to join the junction of the two greatest to that of the two least resistances. If the battery has a greater resistance than the galvanometer, the battery ought to be placed between the junction of the two greatest and that of the two least resistances. * A full discussion of the question is given in The tlieonj and practice of ahiolute meanurementt in EUctricity and MagnetUm^ by Andrew Gray, F.R.8. 288 ELECTRICITY AND MAGNETISM LVIII Thus if the galvanometer has the greater resistance, and if P = 100, Q=10 and ^ = 8 so that iJ = 80, the galvanometer circuit should be connected to B and Ki, and the battery to D and K.2, as shewn in Fig. 108. Whenever P and Q are equal, the best connexion is that in which the battery and galvano- meter circuits are different to that shewn in Fig. 108. Exercise I. Measure to the nearest ohm the given re- sistances. The object of this exercise is to make the student familiar with the connexions of the P.O. Box, and to provide practice in determining a resistance quickly when the highest accuracy is not required. Light the scale lamp, and if the image of the wire on the scale is not sharp make the necessary adjustments. Bring a weak magnet near the galvanometer, and observe the motion of the spot. If the motion is smooth and regular, the needle swings freely. If it is irregular, adjust the galvanometer ac- cording to the instructions in Section LVII. Make connexions to the galvanometer and Leclanche cell as shewn in Fig. 108. Rule columns in your note-book and record the zero reading of the galvanometer, as indicated in the Table below. Make P and Q each equal to 1000 ohms. Take a resistance of 1000 ohms out of the bridge arm AD, and to begin with, leave the resistance to be measured disconnected. Press the key K, to make the battery circuit, then for an instant the key K^ to make the galvanometer circuit, and release first K^, then K^. Observe in which direction the spot of light moves, and note that in the subsequent measurement a deflection in the same direction will always mean that S is larger than R ; i.e. that the resistance in the arm AB is too small. Now connect the resistance to be measured as shewn in the figure, and see that all plugs which are not taken out of the box are firmly in their places, and that all screw contacts are clean and secure. Again press down K^ and then iTj, observing in which direction the galvanometer begins to move. Release iTi quickly, so as to avoid passing a possibly large current through the galvanometer for a longer time than is necessary. If the galvanometer is deflected in the same direction as LVIII MEASUREMENT OF RESISTANCE 289 before, this shews that the resistance to be measured is greater than 1000, because R being 1000, the deflection is in the same direction whether S is equal to the given resistance or infinitely large. In that case a higher value should be tried in the arm AD, say 5000, and so on until a resistance has been found which gives a deflection in the opposite direction. Having thus ascertained by trial that the resistance lies between say 1000 and 5000 some intermediate resistance should be tried, say 3000, and the direction of deflection will then shew between which limits the right value for balance lies. A succession of trials each time halving the resistance approximately will quickly reduce the limits, until two values of R are found differing from each other by one ohm. The one which gives the smallest deflection will give to the nearest ohm the correct resistance of S. Similarly should S have been found to be smaller than 1000, successive trials of 500, 250 etc. will ultimately give a lower limit, and when this is found a successive halving of the interval will again give the required resistance. The chief precaution to be taken by students is to avoid confusion as to the meaning of the defections to one side or the other. As soon as it has been ascertained to which side the deflection takes place, when the resistance taken out is too small, i.e. the direction of motion of the spot of light when S is not inserted in the circuit, this direction must be carefully and con- spicuously noted so that no mistake will aftei^ards be made. The reason for pressing down the keys K^ and K^ suc- cessively is to avoid effects of self-induction, which may cause a quite different distribution of currents at the instant of making the battery circuit. Pressing the key K^ when the connexions are as in the figure completes the circuit, and the first period which is affected by self-induction very quickly dies out, so that Kx can be pressed down almost immediately. Before an approximate balance has been obtained, the galvanometer key should only be pressed down for a sufficient time to shew in which direction the needle moves, as it is desirable to avoid as much as possible the passing of unnece-ssarily large currents through the galvanometer. When the balance is nearly right the key is kept down until the deflection can be read off either by 8. P. 19 290 MAGNETISM AND ELECTRICITY LVIII noting the amplitude of the first swing or by waiting till the needle has come to rest. Determine the resistance of the coils provided and arrange your results as follows : 20 January, 1898. Galvanometer No. 2 (Resistance = 67 ohms), Box C. P=1000. Q=1000. R Zero Deflected reading Deflection 1000 (S=QO) 50-1 to right + 1000 500 200 400 450 420 410 412 413 411 501 50-1 to left >> to right 51-4 46-2 49-2 50-3 50-1 50-0 50-2 + + 1-3 -3-9 - -9 + -2 zero - -1 + -1 .S' = i? = 412ohms. Until a student has obtained sufficient practice, his Book of Observations should contain a statement of all his observa- tions as above. In writing out his results one example should be given in full ; for the other resistances measured, the results only need be given. Exercise II. Determine the same resistances to the greatest accuracy which the apparatus at your disposal will allow. The resistances having been approximately determined, the student should for each of them separately consider the rules mentioned at the beginning of this section and fix on the values of P and Q which he considers most suitable. The rule about the proper connexion of galvanometer and battery should also LVIII MEASUREMENT OF RESISTANCE 291 be attended to. When the galvanometer and battery circuits are interchanged, the order in which the keys K^ and if, are pressed down must of course be reversed also. From his previous results he will at once be able to take out of the arm AD the plugs necessary to adjust the balance almost correctly. A few further trials, if necessary, will then again lead him to the two values R and ii + 1 between which the balance lies. The deflections for both these must then be carefully observed on the scale, and by interpolation one, and possibly two decimals may be found. If the deflections are not sufficiently large, the galvanometer must be made more sensitive. Resistances of not less than five ohms should in each case be accurately determined to -1 */„ and if possible more accurately. To ensure this accuracy, however, with the smaller resistances it will be necessary to attend carefully to the connexions and to make sure that the plugs in the resistance box are firmly in their sockets. If the resistances are of copper or some other unalloyed metal, the resistance of which increases about 1 7o for every 3° C, the temperature should be noted, and the current should be passed through the coil for as short a time as possible, so as to avoid heating effects, otherwise the resistance will be found to increase gradually. To make sure that no such change is taking place, the observations should be repeated. Owing to the temperature effect, the accuracy obtainable with pure metals is much smaller than with alloys, which have a small temperature coefficient. Enter your results as follows : Material P Q Junction between Pand Q connected to R Temp, of coil Coil No. 1 2 Copper Platinoid Manganin 1000 100 100 1000 10 10 Galvanometer Battery Battery 412-3 68-25 7-423 16'-2 16'-4 16--9 19—2 292 MAGNETISM AND ELECTRICITY LVIII Exercise III. To determine the resistivity of the material of the wire. Measure the resistance of the wire provided to 'lYo, noting the length of the wire under the screws which clamp it to the resistance box. Measure the diameter of the wire in four places and the total length. Subtract the length under the clamping screws from the total length, and calculate the resistivity. Arrange your results as follows : 21 January, 1898. Resistivity of Platinoid. Total length of wire = 100-2 cms. Length under screws ... = 1-7 „ Length (I) used ... = 98-5 „ Diameters at different places screw gauge Mean diameter ... by. = -0.342, -0344, -03- = -0345 cms. Mean cross section (a) ... ... = -000937 sq. cms. Resistance (R) ... = 4-205 ohms. Resistivity = -r- ... = 40-0 X 10-« ohm SECTION LIX. MEASUREMENT OF A HIGH RESISTANCE. Apparatus required : High resistances, resistance boxes, cells and mirror galvanometer. When the resistance to be measured, is large compared to that of the most sensitive available galvanometer, the Wheatstone bridge loses its advantages and ceases to give more accurate results than simpler and more direct methods. When the accuracy required does not exceed about *! 7o these simpler methods will be sufficient. If a standard resistance of approximately the same value as the one to be measured is available, the method known as the " direct deflection method " may be used. Let a cell of resistance B ohms be connected in series with a high resistance of R^ ohms, and a mirror galvanometer of resistance G ohms (Fig. 109). The current C^ through the circuit is given by C- E Similarly when a second known resistance R, is substituted forR,, C- g so that B + + R,-^'(B + + R,). 294 MAGNETISM AND ELECTRICITY LIX The battery resistance will in nearly all cases be quite negligi\)le compared to the resistance to be measured, and the resistance of the galvan<>meter will be known or must be determined by an independent measurement. Hence R^ being known, R^ can be calculated from the above equation, if the deflections of the galvanometer d^ and d^ corresponding to Ci and Cg are observed, and the ratio of currents deduced from the ratio of deflections. If the indications of the mirror galvanometer have been calibrated (page 281) a Table is probably available in the laboratory by means of which de- flections may be converted into currents. In the following exercise it is assumed that results of sufficient accuracy may be obtained by assuming the deflections to be proportional to the currents. In that case the unknown resistance R^ is given by and if G is small compared to R^ and R2 Taking a resistance box having a total resistance of not less than 10000 ohms as R^, determine the re- sistance Rz of the coil provided. Arrange the . — (^ circuit as in Fig. 110, the resistances to be compared being placed side by side in such a way, that either one or the other may be put in ' .' circuit with the battery and galvanometer. If the deflections of the galvanometer are too large to enable observations to be taken, diminish the sensitiveness of the instrument by an external magnet. If this is not possible arrange the circuit as in the second method described below. If the resistance to be determined is more than three or four times that of the standard, the ratio of the currents will not be capable of measurement to a sufficient degree of accuracy by the direct deflections. The method should then be modified by the addition of an arrangement which allows the electro- _Lb LIX MEASUREMENT OF A HIGH RESISTANCE 295 motive force to be varied in such a way that the deflections di and (L are not very unequal. For this purpose a cell is connected to the ends of a resistance box containing about 1000 ohms. The galvanometer with high resistance to be measured, is connected to part of the resistances only (Fig. Ill), and this part may be varied. Ivfy Let the resistance to be measured be R^ — lisp-v and the standard resistance ii,, and write M fi, H^ S,^R, + G, S, = R,-¥G, Fig. 111. where G is the galvanometer resistance. Let r be the total resistance inserted in the box, and ri, r^ that part of it, which lies between the terminals of the galva- nometer circuit according as R^ or R^ is included, the currents through the galvanometer being in these cases Ci or C^ ; then if E is the electromotive force of the battery, the resistance of which may be neglected compared to the other resistances in the circuit, a simple application of the laws of derived circuits gives C,(^,r + (r-rOn) = AV„ C, {S^r + (?• - ra) ra) = Er^, from which we find C^r,\ r J r Q Hence R^ may be determined, if we can tind the ratio 77 from the galvanometer deflections. If the currents are not pro- portional to the deflections or to their tangents, the galvanometer must be calibrated. But this can be avoided and the arith- metical work shortened if the resistances r,, r^ are so adjusted that the deflections are equal. In that case The above equation also simplifies if the total resifltftDoe is 296 MAGNETISM AND ELECTRICITY LIX inserted, when the larger of the two resistances R^ and R^ is in circuit. In that case when r^=-r, we have ^-S(¥-^(--)) and when 7-1 = r, we have Should it be necessary to take the battery resistance into account, we need only add its value to that of r in the complete equation. Determine by this method the resistances of the blacklead line, and of the samples of insulating material provided. From the dimensions of the samples, calculate the resistivities of the different materials. Assuming the deflections to be proportional to the currents, tabulate your observations and results as follows : Direct Deflection Method. 20 January, 1898. Galvanometer No. 2 (Resistance 6,430 ohms). Standard Resistance (R^) = 10,000 ohms. Besistance Deflection 10000 10000 508-7 274-5 510-3 d,= 509-5 d,= 274-5 R, = 16430 X 2^^ - 6430 = 24,066. LIX MEASUREMENT OF A HIGH RESISTANCE Shunt Method. 20 January, 1898. Galvanometer No. 2 (Resistance 6,430 ohms). Standard Resistance = 25,000 ohms. r= 11,000 „ 297 Besbtanoe ri Deflection 25000 Pencil Line 25000 1000 11000 1000 430 489 436 rf, = 433 ri= lOuO 6', = 31430 d^ = iS9 r, = r= 11000 4,00 S, = *^^ {(31430 X 11) + 10000) = 314,900, 222=208,500 ohms. Similarly for the samples of insulating material. SECTION LX. MEASUREMENT OF A LOW RESISTANCE. Apparatus required : Low resistances and standard ad- justable resistance^ Daniell cell, two four-way keys, and mirror galvanometer. When a resistance which is only a small fraction of an ohm is to be measured, the Wheatstone bridge is no longer capable of giving accurate results owing to the resistances introduced at the various contacts. Several special methods have been devised, one of which is known as the *' fall of potential method." It depends on the comparison of the differences of potential produced by the same current at the ends of the unknown resistance and of a known standard resistance. The two resistances R^ and R.^ (Fig. 112) are connected in series with a third resistance i2 of a few- ohms, introduced to regulate the current, and to a cell B giving a constant electro- motive force, e.g., a Daniell. From the ends of the two resistances R^ and R^, wires are taken to keys which enable each resistance to be placed in parallel with a galvanometer G, the resistance of which is ^10,. 112. high compared to the resistances under test. When this is the case, the currents in the galvanometer will be proportional to the differences of potential at the ends of the resistances, which will in their turn be proportional to the resistances themselves. The galvanometer should give de- flections sensibly proportional to the currents passing through them, or if this is not the case, its indications should have been previously calibrated. LX MEASUREMENT OF A LOW RESISTANCE 299 If i^a is a standard resistance not many times greater or many times smaller than i^, the latter may be determined by this method with an accuracy which is generally sufficient. The connexions to tiie galvanometer may be arranged by means of two four- way keys, K^ and K^ as indicated in the figure, the plugs being inserted to connect the galvanometer to the two resistances in turn. After making connexions proceed as follows : Make i2=10 ohms, connect the galvanometer to R^, and read the deflection. If the detlection is so great that the spot of light leaves the scale, increase R. Connect the galvanometer to ilj and again read the deflection. If the larger of the two deflections is not at least half way across the scale, connect the galvanometer to the resistance which gives the larger deflection, and diminish R till this deflection is as large as can conveniently be read. Determine it accurately. Transfer the galvanometer terminals to the other resistance and read the deflection, then return to the former and again read the deflection. If any measurable change has occurred take two more observations and record as shewn below. In the reductions, assume that the galvanometer deflections are proportional to the currents. Enter your observations as follows : 10 June, 1899. Galvanometer No. 3. Resistances marked A, B, G. Galv. connected up to Deflections, scale divisions Mean 114-5 29-6 ^.W J?, = 01 ohm 114 ... 115 29-4 ... 29-8 114-5 H( R, 114-5 = ;}-87 29-6 or resistance A = '0387 ohm. Similarly for the other given resistances. SECTION LXI. MEASUREMENT OF THE RESISTANCE OF A GALVANOMETER. KELVIN'S METHOD. Apparatus required : Galvanometer, Post Office resistance box, Leclanche cell, resistance box of 1000 ohms and connecting wires. In the ordinary Wheatstone bridge arrangement, if the four arms of the bridge (Fig. 108, p. 285) satisfy the relation F/Q = R/S there is no difference of potential between the points A and B, and consequently if these points be connected by a wire, no current will pass through it. Such a wire would therefore not alter the strength of the currents in any of the branches of the bridge and on the other hand we may conclude that the above relation is satisfied, if the current in any one of the branches is the same whether A and B are directly connected together or not. If the branch R contains a galvanometer, the deflection of which serves to indicate the strength of the current through it, we may judge whether the resistances are balanced or not by making and breaking the contact of a wire connecting A and B. If the deflection is the same in both cases the resistance in the branch R must be equal to PS/Q and may therefore be determined without the assistance of a second galvanometer between A and B. The above considerations furnish an interesting and useful method to determine the resistance of a galvanometer. To obtain accurate results it is in general necessary to send a current through the instrument, which would under ordinary circumstances drive the spot of light off the scale. Reducing LXI RESISTANCE OF A GALVANOMETER 301 the sensitiveness of the galvanometer so as to make the deflection measurable would not get over the difficulty, because the test itself would become less sensitive. It is possible however to work with a large deflection and yet have the spot of light on the scale, because the zero reading of the galvano- meter is not required and may therefore be outside the limits of the scale. In the case of a galvanometer of the suspended needle type it is convenient to begin by deflecting the needle with the help of a weak magnet, which should turn the field rather than increase its strength, and the deflection should not be too large to begin with, so as to save passing currents through the galvanometer, which may cause damage. If the battery is then connected through a variable resistance, as shewn in Fig. 118, so that only a portion of the current passes through the bridge, the spot of light is brought to the scale by variation of the resistances r and r^. Care is necessary of course to send the current in the direction necessary to bring the suspended system into the required position. The method adapts itself very easily to the measurement of the resistance of a D'Arsonval galvanometer, for it will only be necessary to rotate the whole instrument through an angle of about 30° and to adjust r and 1\, until the deflection can be read on the scale. The best arrangement of the bridge, when Q may be made smaller than the resistance of the galvanometer, is that in which Q is as small as possible and P as large as possible without R being caused to exceed the maximum resistance available in the arm AD. The battery should connect the points A and By i.e., the battery and short circuiting wire in the figure should be interchanged. When the lowest available resistance of Q is larger than the galvanometer resistance, Q and P should be made equal to each other and as large as possible, and the battery should then be connected as shewn in the figure. If the resistance of the galvanometer is not even approximately known the last-mentioned arrangement is the most suitable and the sensitiveness will generally be sufficient In the following exercise, it is required to measure the resistance of a d'Arsonval galvanometer, the method for other 302 MAGNETISM AND ELKCTRICITF LXI Fig. 113. galvanometers only differing as explained above in the means adopted to alter the zero reading. Make connexions as shewn in the figure. With rj small and r large, pass a current of short duration through the galvanometer by pressing down the key K^ instantaneously. Turn the galvanometer through an angle of about 30" or 40° in the opposite direction to that in which the spot of light moved on making contact. With F=^Q = R = 1000 close the bat- tery circuit at the key K^ and adjust 7\ and r until the spot of light is on the scale. This is done most rapidly by watching the galvanometer mirror directly. Press the short circuiting key K^ and observe the side to which the spot of light moves. Then make R = 5000 and repeat the observation. If the motion is in the same direction as the first it indicates that S lies below 1000 or above 5000. The first supposition being the more probable in .the present case, repeat the observation with i^ = 500. Proceed as in Section LVIII to find the value of B for which the spot of light does not move when the key K^ is closed. When the first deflection of the needle is sufficiently small, take readings of the first swing and find the value of R for accurate balance, if necessary by interpolation. If the resistance of the galvanometer is greater than the lowest available resistance of Q, which will be 10 with the resistance boxes in common use, interchange the battery and short circuiting wire. Make Q=10 and P = 1000 if a resistance of 100 >S^ can be placed in the arm AD, otherwise make P= 100. Take out of the branch AD the resistance which according to the previous determination should produce balance; and if there is a deflection proceed to improve your result. When the balance is nearly perfect wait a few minutes in order to allow the galvanometer coil to take up the temperature of the room, it having probably been heated by the passage of currents sufficiently to affect its resistance sensibly. LXI RESISTANCE OF A GALVANOMETER Enter your results as follows : 9 June, 1899. Galvanometer D' Arson val C. Battery joining C and D. P = 1000, Q=1000. 303 R Beading with JBopen Beading with AB closed Deflection 1000 15-1 + large to right 5000 121 + jj » 500 23-2 + j> >> 100 25-4 — „ to left 1 300 240 41-3 + 17-3 200 240 22-5 - 1-5 220 23-6 24-5 + -9 210 23-4 22-4 - 10 Resistance of galvanometer = 215 app. Temperature of room = 15°'4. The battery and short circuit were now interchanged and with Q=10, P = 100 it was found that P = 2154 gave a balance but that this resistance seemed slowly to increase owing no doubt to the heating of the current. The same result was obtained after waiting five minutes. Hence for the final result : Resistance of galvanometer = 215*4 Temperature = 15°'4. SECTION LXII. DETERMINATION OF THE RESISTANCE OF A CELL. lodge's modification of mange's method. Apparatus required : Daniell and LeclanchS cells, Post Office resistance box, high resistance galvanometer. If in the Wheatstone bridge arrangements (Fig. 108, p. 285) the relation PS = QR is satisfied, the portions of the circuit CD and AB are said to be conjugate to each other. No electro- motive force in one of these portions will produce a current in the other, and no change of resistance in one of them will modify any current in the other, such as may be produced by an electromotive force in one of the branches of the bridge. This fact was first used by Mance to determine the resistance of a cell. He placed the cell in one of the branches, and replaced the battery which in the ordinary Wheatstone bridge supplies the current, by a simple key iL'2(Fig. 114). Owing to the presence of the cell in BD Fig. 114. a current passes through the galvano- meter, but when PjQ — R/S this current should be the same whether the contact at K^ is open or closed. Mance's method in its original form has several disadvan- tages. A comparatively large current has to be sent through the galvanometer and the key K^ has to be pressed down, at least for the length of time equal to one quarter of the period of the galvanometer suspension, so as to allow the first deflection to be read. This may affect the galvanometer injuriously, and it also produces disturbing effects in the cell itself The closing of the circuit at K^, though it does not affect the current in the LXII DETERMINATION OF THE RESISTANCE OF A CELL 305 galvanometer branch, will increase the current through the cell itself, and this increase is accompanied by changes in the resist- ance and electromotive force of the cell. This is a disturbing cause which affects all measurements of the resistance of a cell, but the difficulty may to some extent be overcome by allowing the key K^ to be closed only for a very short interval of time. The modification of Mance's method, introduced by Lodge, gets over the difficulty by the introduction of a condenser in series with the galvanometer (Fig. 115). The condenser will be charged to a difference of potential equal to that between the points A and B\ and whenever a sudden change in that difference of potential occurs, an instantaneous current will pass through the condenser. When however the relation PjQ — RIS is satisfied, the galvanometer will not be affected by the sudden pressing down of the key K^. The most sensitive arrangement for the determination of the resistance of a cell is that, in which Q is equal to the resistance of the cell and P is as large as possible. The connexions should be as in Fig. 115, the short circuiting key being in the branch CD. It is more important that Q should be nearly equal to the cell resistance, than that P should be large. If P = lOQ the arrangement has 90 7© of the greatest possible sensitiveness, but when Q=^\OS the sensitiveness is reduced in the ratio of 10 to 1. Hence when the cell has a very low resistance, as e.g. a storage cell, either some special arrangement should be made to reduce the value of Q, or an accurately known resistance should be placed in series with the cell, so that the total resistance of *S' is not less than half that of the lowest available value of Q. Set up a Daniell cell, taking care that the zinc is not covered with a deposit of copper. Place it in the arm BD of the Wheatstone Bridge Box ABCD (Fig. 115), putting a plug key K in circuit, in order that the current can be stopped except when observations are being taken. 8. p. 20 306 MAGNETISM AND ELECTRICITY LXII Place a condenser T of about "3 microfarad capacity in series with a high resistance galvanometer, and connect to the bridge as shewn. Make Q=10, P = 1000, and choose for first trial a resistance R which you expect to be greater than that of the cell to be tested, viz. 100 in the present instance. Make circuit at K^y then momentarily at K^. Note the direction in which the spot of light moves. Reduce the value of R and proceed as in Section LVIII. to determine its value for balance. If the key K. is held down, the spot of light will be seen to drift owing to changes in the cell, brought about by the passage of the current. Wait five minutes without passing a current through the cell, and determine the resistance again. Repeat again after further five minutes. Determine the resistance of the Leclanche cell in the same way, then placing the two cells first in series, then in parallel, determine their joint resistance. Record as follows : 6 February, 1899. P=1000, Q= 10 ohms. 1. Daniell cell, No. 3. Balance obtained with R = 515, i.e. S = 5 15 ohms. After five minutes ... ... ... =5*10 „ After further five minutes ... ... =5*07 „ 2. Leclanche cell, No. 4. Approximate balance obtained for R between 470 and 480 S = 4*75 approx. After five minutes and after a further five minutes, the resistance was found approximately the same. The resistance therefore varies be- tween 4*7 and 48 ohms. 3. Leclanche and Daniell cells. (a) in series ... ... ... ... S = 8'9 ohms. (6) in parallel =22 „ Calculated resistance case (a) ... = 98 „ case (6) ... =25 „ SECTION LXIII. COMPARISON OF RESISTANCE STANDARDS. CAREY FOSTERS METHOD. Apparatus required : Carey Fosters bridge with sioitch, two equal resistances, standai^ds to be compared^ miiTor galvano- meter, voltaic cell. When nearly equal resistances, as for instance a number of ohm standards, are to be compared together with gi-eat accui-acy, a modification of the bridge method due to Carey Foster may be used. The two "proportional arms" P, Q (Fig. 116) of the bridge are made as nearly as possible equal, and the two standards R, S to be compared together with a straight wire CC join- ing them form the other arms. The standards R, S are connected to the bridge in such a way that they can be readily interchanged by means of a switch having mercury contacts. The wire CC is stretched along a gradu- ated scale, on which the position of the point of contact of the sliding connexion leading to the galvanometer may be read off. Let a balance be found when the galvanometer contact is made at a point reading x along CC\ R and S being placed as shewn in the figure, and let x' be the position for a balance when R and 8 are interchanged. Let p be the resistance 20—2 308 MAGNETISM AND ELECTRICITY LXIII of a length of wire GC equal to one scale division, then since the resistance from B to D through CC remains the same on interchanging R and S, the sum of the resistances BL and LD remain the same ; and since when there is a balance their ratio must also be the same, it follows that the resistance from Bio L is the same before and after the change. Hence R + px = S-\-px or R — 8 = p(x— x). If p is known the difference between R and S can therefore be calculated. To obtain p diminish the effective resistance of S by placing a large known resistance Si in parallel with it. Balance, interchange the two arms, and balance again. Let x^ and Xi be the two readings. Then ~ SVS ^P^^' ~ ^'^' Eliminating R we obtain ^_S^ 1 S + Si (xi — Xi) — (x — x) ' It is here assumed that the bridge wire is uniform, so that its resistance is strictly proportional to its length. When the highest accuracy is required, the wire should be calibrated so as to correct for any inequality either of the cross-section or the material of the wire. See that the mercury contacts of the switch are clean. Place the terminals of the two equal coils, which are best wound on the same bobbin, and those of the two coils to be compared in the mercury cups provided for them in the mercury switch. Connect the Leclanch^ cell through a key, and the galvanometer to the bridge, attending to rule (3), page 287. Place a thermo- meter in the centre tube of each coil, and when the indications have become steady, read the temperatures. Obtain a balance and note the reading of the sliding contact. Reverse the con- nexions of the standard coils by moving the switch, balance again and note the reading. Read the thermometer again. Place, in parallel with S, a high resistance of amount sufficient LXIIl COMPARISON OF RESISTANCE STANDARDS 309 to displace the point of balance by about a fourth of the length of the wire, and determine the position for balance. Again reverse and balance. Read the temperatures. Record as follows : 2 October, 1900. Wolff's standard 1 ohm (R) compared with Hartmann and Braun's standard 1 ohm (S). Temperature coefficient of R said to be zero ,, ^=-00020 S according to the certificate is correct at 18'''2. Balance with R on left side of bridge ... x = 24*38 cms. „ „ o „ „ „ „ .,. X = ^4"7o „ .-. R^S='S7p, t = 15°'lo. 100 ohms were placed in parallel with S and difference in readings now found to be 4*52 cms. Hence p^^^.^^^-^ = ^^ A^ = 00239 ohms per cm. 419'2 ^ .-. R-S^002S9 X -37 = -00088 ohms. Resistance of S at 15°-2 = 1 - 00060 = -99940 ohms, of ie at 1 5°-2 = 1 00028 ohms. SECTION LXIY. CHANGE OF ELECTRICAL RESISTANCE WITH TEMPERATURE. Apparatus required : Two coils of luire in a tube which can be raised in temperature, Post Office bridge, voltaic cell, and mirror galvanometer. The electrical resistance of a wire of a pure metal increases rapidly with increase of temperature, while that of a wire of an alloy increases more slowly, and by a proper choice of the con- stituents, may be made to remain nearly constant at ordinary temperatures. To determine the change of resistance of a wire due to temperature, the wire may be wound round a sheet of mica and placed, along with a thermometer for indicating its tem- perature, in a test tube surrounded by water which can be heated. The apparatus supplied is constructed on this principle. It consists of a brass vessel, in which is placed a test tube containing two coils, one of copper and one of platinoid wire, immersed in petroleum and joined to the three screws on the wooden disc through which the tube passes, in such a way that the middle screw is connected to one end of both coils, while the other ends of the coils are separately connected to the other two screws. Fill the brass vessel full of tap water, and place the disc through which the test tube passes over the vessel, so that the tube is immersed in the water (Fig. 117). Connect the common terminal of the two coils to a Post Office Wheatstone LXIV CHANGE OF RESISTANCE WITH TEMPERATURE 311 bridge in the usual way, and the other terminals to two of the screws of a three-way key, the third screw of which is connected to the other bridge ter- minal, and determine the resistance of each coil at the temperature of the bath. The galvanometer should be sufficiently sensitive to allow the thousandth part of an ohm to be estimated, and the temperature readings should be correct to '05° C. Rixise the temperature of the water to about 20° C, and keep it at that temperature till the reading of the thermometer in the test tube is steady, then repeat the obser- vations of the resistances of the two coils. Take further observations at about 30°, 40°, 50°, 60°, then cool the water, and redeter- mine the resistances at the same temperatures (within about a degree) as previously. Record as follows: 12 December, 1898. Post Office Box C. Coils No. 3. Fig. 117. Copper CoU. Platinoid CoU. Temp. C. Resist, olims ' Temp. 1 C. i Resist, obms Temp. Resist. Temp. Resist. 13'-62 20 15 29-80 39 15 49-50 59 -90 4'-168 4 -274 4 -418 4 -552 4-727 4-915 13" -58 20 05 29 -90 39 -35 49 -60 1 59 -70 1 4162 4-258 4-434 4-584 4-743 4-881 j Draw curves representing the results, taking temperatures 08 abscissae and resistances as ordinates. It will be found that for the copper coil the curve is almost a straight line, which, when continued so as to take in tem- peratures below the freezing point, gives zero resistance at the zero of absolute temperature. 312 MAGNETISM AND ELECTRICITY LXIV All pure metals behave in the same way but the resistance of alloys follows other laws. The resistance of a pure metal may be represented very accurately by the equation ?V = ?'o(l + a^ + /3«'), where Vt is the resistance at f and r^ at 0°, and a and /3 are constants depending on the nature and state of the metal. The temperature coefficient of increase of electrical re- sistance at f is the increase in resistance of a conductor for a rise of temperature from {t — \y to (^ + i)° divided by the resistance of the conductor at 0°C. It is therefore in the above case a + ^(St The quantity y8 is small compared to a : thus for platinum a = + 003448, yS = - 0-000000533. For many purposes it will be sufficient to neglect /9, and to take a to be the mean temperature coefficient for the range of temperature considered. In that case, writing R^ and R.^ for the resistances at the temperatures t^ and 4, we have R^^r,{\+at,), R^ = r,{l+OLU), and by elimination of r^ !._ Jl Xt2 — -til -^1 R^ jRa Ri The first formula involves rather less calculation than the second, but the second allows a more symmetrical arrangement of the observations and calculations, and as tables of reciprocals should be available in every laboratory, the introduction of con- ductances in place of resistances involves little additional labour. Calculate the temperature coefficient of the electrical re- sistance of copper by taking the mean of each pair of nearly equal temperatures in the first and third columns of the above Table and the mean of the resistances found for these tempera- tures. Arrange them in two sets and carry out the work as follows: LXIV CHANGE OF RESISTANCE WITH TEMPERATURE 12 December, 1898. Copper. Coil 3. 313 MeM Mean re«Ut. Ohms. 1 R ( ^ Meui temp. Cent. Mem reebt Ohms. 1 R t R Diff. of 1 R Diff. of t R lUtio 13-60° 2010 29-85 4-165 4-266 4-426 •2401 •2344 •2259 3-265 4-711 6-744 39-25° 49-55 59-80 4-568 4-735 4-898 •2189 •2112 -2042 8-591 10-466 12-211 •0212 -0282 •0217 5-326 5-755 6-467 •00398 403 397 a = mean = •00399 Reduce the results for the platinoid coil in the same way and examine whether in its case the above simple relation between temperature and resistance holds. Note on Platinum Thermometry. The measurement of change of resistance due to change of temperature has become of considerable importance, since it has been found that it may serve as a basis for the measurement of temperature. One of the advantages of this method is, that by the use of a metal with a high fusing point, e.g. platinum, it may be applied to tempera- tures which are so high that mercury or air thermometers cannot be used. A platinum thermometer must agree with the ordinary scale at the fundamental points 0° C. and 100° C, and this may be done by defining the temperature as measured by the platinum thermometer {tp) to be ^^=100-?---^ where r,, r^, r,oo are the resistances of a platinum wire at 0°, p", and lOO"* respectively. This definition makes the rise of temperature proportional to the increase in the resistance, as is seen by writing the difference between two temperatures tp and tq in the form <p-<,= 100 ^ifflo ~~' rn The definition secures also that when rp is equal r^ or Tim the temperature indicated shall be or 100 respectively. 314 MAGNETISM AND ELECTRICITY LXIV The difference between a platinum and an air thermometer at any temperature can be calculated, if the change of resist- ance of platinum as depending on the indications of the air thermometer is known. Thus putting rp = ro(l+a« + W, where t is the temperature as measured in an air thermometer, we find by substitution ^ en- 100^ This equation was first used by H. L. Callendar, who has measured the quantities a and /8 for pure platinum with great care and found the values given, p. 312. With these values the difference of the two temperature scales becomes finally ^-.^^ = -0-157^ (100-0. The numerical coefficient has however to be redetermined for each sample of platinum. SECTION LXV. MEASUREMENT OF THE RESISTANCE OF ELECTROLYTES. Apparatus required : Post Office resistance box, electrolyte tube, telephone, induction coil, storage cell, and microscope. The resistance of an electrolyte cannot be measured by placing the vessel containing the liquid in the Wheatstone bridge in the ordinary way, since the passage of the current produces polarisation at the electrodes, and therefore sets up an electromotive force, which has the same effect on the measuring instrument as a change of resistance. If however an alternating instead of a direct current is sent through the bridge, the polarisation due to the passage of the current in one direction, is neutralised by the passage the next instant of an equal current in the opposite direction, and if the changes succeed each other with sufficient rapidity no appreciable effect on the measurement is produced. The galvanometer must however be replaced by an instrument capable of detecting alternating currents, as for example an electrodynamometer or a telephone. The latter is more generally used at present. The use of alternating currents necessitates care in avoiding appreciable self-induction and capacity in the resistances, since the Wheat- stone bridge does not measure in that case "resistance" but " impedance," which depends on the self-induction, the capacity and the number of alternations per second as well 8is on the resistance. It is owing to the impossibility of completely getting rid of or balancing self-induction and capacity, that no perfect balance is ever obtained unless the alternating currents used follow the law of sines. 316 MAGNETISM AND ELECTRICITY LXV Important conclusions on the molecular constitution of solutions have been drawn from their electric conductivities, the solutions of different substances being compared when their concentration is such, that an equal number of molecules are dissolved in each case. If the number of atoms in one gram of hydrogen be 91, and a represent the molecular weight of a substance referred to hydrogen, then a grams of any substance will contain the same number n of molecules, and different solutions each containing a grams per litre will therefore be comparable with each other. The mass of a substance which contains as many grams as is indicated by its molecular weight, is called a " gram-molecule," an expression which, it is hoped, will before long be replaced by a more appropriate one. A solution which contains one gram-molecule of a salt per litre of water is called a "normal solution." The molecular conductivity of an electrolyte is the conductivity divided by the number of gram-molecules which are dissolved in one cubic centimetre. If it is required to measure the conductivity, not only relatively to some standard solution but absolutely, the electrolyte must be inclosed in a tube of known length (I) and cross-section (a). If R is the resistance of such a tube Ra/l will be the resistivity, and l/aR the conductivity. If the tube is conical, a^ and Og being the cross-sections at the ends, it will have a resistance equal to that of a uniform tube having a cross-section equal to s/a^a^. If each cross -section is circular, it is calculated in the usual way from the diameter, but if it is elliptical in shape, the two prin- cipal diameters d^, d^ must be measured, and the area is then equal to ^irdyd^. If d^, d4 represent the principal diameters at the other end, the cross-section of the equivalent uniform tube will therefore be \'Tr\fd^d^Qd^. In general the four diameters will be nearly equal, and if they do not differ by more than 2 or 3 per cent, it will be sufficient to substitute arithmetical for geometrical means, so that ii D = \{d^ + d^-\- d^ + d^) the area to be used in the reduction of the experiment will be ^irD'^. Wash out the narrow glass tube and wider end tubes provided, and fill them with a solution of sodium chloride of four times normal strength, i.e. containing 4 gram-molecules per litre, place the platinum electrodes in the end tubes and LXV RESISTANCE OF ELECTROLYTES 817 connect to the bridge as shewn in Fig. 118, taking out the 1000 ohm plugs from each arm of the box before connecting up. Fig. 118. Place the bulb of a thermometer in one of the end tubes, taking care that it does not come between the platinum electrode and the end of the narrow tube. To the points C, D of the bridge to which the battery is usually connected, join the terminals of the secondary of a small induction coil /, the primary of which is excited by a cell E of sufficient power to work the coil. Connect a telephone T to the terminals A, B. Determine the resistance in the adjustable arm of the bridge to produce a minimum sound in the telephone. Take about 50 c.c. of the solution and dilute to double the volume, i.e. make a solution of twice normal strength, and after washing out the tube, fill it with the new solution and deter- mine the resistance. Dilute down to normal, then to half, and quarter normal strength, determining the resistance and observing the temperature in each aise. Remove the end tubes of the electrolyte cell, and measure the internal diameter of each end under the microscope. If the tube is not quite circular measure the least and greatest diameters of each end. Measure the length I of the tube. 318 MAGNETISM AND ELECTRICITY LXV Arrange your observations and results as follows : 12 December, 1898. Electrolyte tube A. Resistance box C. Microscope A. 75*4 eyepiece divisions = 1 cm. of stage scale, .-.1 „ „ =-0133 cm. Diameters = 236, 232, 234, 234 eyepiece divisions. Mean Diameter = 23*4 eyepiece divisions, = •311 cms. Area = ?^. (-311)2 Length of tube Z = 129 cms. a •0759. = •OOoSS. Sodium chloride solution at 18° C. Strength Kesistance Resistivity Conductivity Molecular Conductivity 4 normal 840 ohms 4-94 •202 50-5 2 „ 1290 „ 7-59 •132 66 normal 2235 „ 13-1 •076 76 \ normal 4230 „ 24-9 •040 80 "k " 8060 „ 47-4 •021 84 Draw a curve shewing the relation of conductivity to strength. By placing the tube in a water-bath, the influence of tem- perature on the conductivity of an electrolyte may be found, and expressed by a curve with temperatures as abscissae and conductivities as ordinates. SECTION LXVI. CONSTRUCTION OF A CLARK STANDARD CELL, Apparatus required: H tube and stand, clean mercury, mercurous sulphate, pure zinc, zinc oxide, zinc sulphate, re- agents, paraffin and corks. In one form of the Clark standard cell the active materials are enclosed in a H-shaped tube (Fig. 119), through the lowest Cork Paraffin ZnSO^ cryg Hg,SO,r Hgr I ZDSO4 solution ZnSO^ crystals ZnHg Fig. 119. points of which the platinum wires forming the terminals of the cell pass. The wire on the left (Fig. 119) ends within the tube in a small quantity of pure mercury, and that on the right in zinc amalgam. The mercury is covered with a paste of mercurous sulphate, and this again with a few crystals of zinc sulphate. The amalgam is also covered with cr^'stals, and the rest of the tube tilled with a saturated solution of pure zinc sulphate. Both tubes are closed by cemented corks. The following instructions should be carefully attended to. 320 MAGNETISM AND ELECTRICITY LXVI 1. To secure purity of the mercury it should be first shaken up in a bottle with dilute hydrochloric acid and then distilled in vacuo. 2. To prepare the zinc amalgam add 4 grams of '* com- mercially pure " zinc to 36 grams of the mercury, and heat to 100'' C. in an evaporating dish on a water-bath. If the surface of the zinc is clean it quickly becomes amalgamated, and the zinc slowly dissolves in the mercury, which should be occasionally stirred. If the amalgamation does not occur immediately, the zinc should be removed, treated with dilute hydrochloric acid, dried and replaced. The amalgam should be liquid at 100° C. and solid at ordinary temperatures. 3. Mix in a flask 40 grams of distilled water with nearly twice its weight of crystals of '* pure recrystallised " zinc sulphate, and add about two per cent, of zinc oxide to neutralize any free acid. Heat gently, never allowing the temperature to exceed 30° C. When the crystals have dissolved add about 10 grams of mercurous sulphate treated as described in 4. Filter the solution while still warm into a stock bottle. Crystals should form as it cools. The object to be attained is the preparation of a neutral solution of pure zinc sulphate saturated with ZnSOj + 7iifi. At temperatures above 30° C. the zinc sulphate may crystallise out in another form ; to avoid this 30° C. should be the utmost limit of temperature. At this temperature water dissolves about 1'9 times its weight of the crystals. If any crystals remain undissolved they are removed by the filtration. The amount of zinc oxide required depends on the acidity of the solution, but 2 per cent, will be ample in all cases likely to arise in practice with reasonably good zinc sulphate. Another method would be to add the zinc oxide gradually until the solution became slightly milky. The solution, when put into the cell, should not contain any free zinc oxide; if it does, zinc sulphate and mercurous oxide are formed on introducing the mercurous sulphate, and the oxide may deposit on the zinc, and affect the electromotive force of the cell. The difficulty is avoided by adding, as described, the 10 grams of mercurous sulphate before filtration; this is more than LXVI CONSTRUCTION OF A CLARK STANDARD CELL 321 sufficient to combine with the whole of the zinc oxide origi- nally put in, if it remained free. The mercurous oxide formed, and any undissolved mercurous sulphate, are removed by filtration. 4. Take about 20 grams of mercurous sulphate purchased as pure, and wash it thoroughly with cold distilled water by agitation in a bottle ; drain off the water and repeat the process at least twice. After the last washing, drain off as much of the water as possible. Mix the washed mercurous sulphate with a little zinc sulphate solution, and about half its weight of pure mercury, adding sufficient crystals of zinc sulphate from the stock bottle to ensure saturation. Shake these well up together to form a paste of the consistency of cream. Heat the paste for an hour to a temperature not exceeding 30° C, agitating it from time to time, then allow it to cool, shaking it occasionally while it is cooling. Crystals of zinc sulphate should then be distinctly visible, distributed throughout the mass; if this is not the case, add more crystals from the stock bottle, and repeat the whole process. This ensures the formation of a saturated solution of zinc and mercurous sul- phates in water. The above treatment of the mercurous sulphate has for its object the removal of any mercuric sulphate, which may be present as an impurity. Mercuric sulphate decomposes in the presence of water into an acid and a basic sulphate. The latter is a yellow substance — turpeth mineral — practically insoluble in water; its presence, at any rate in moderate quantities, has no evil effect. If, however, it is formed, the acid sulphate is also formed. This is soluble in water and the acid produced affects the electromotive force of the cell. The object of the washings is to dissolve and remove the acid sulphate, and for this purpose the three washings described will in nearly all cases suffice. If, however, a great deal of turpeth mineral is formed, it shews that there is a great deal of the acid sulphate present, and it will then be wiser to obtain a fresh sample of mercurous sulphate rather than to try by repeated washings to get rid of all the acid. The free mercury helps in the process of decomposing the acid salt, forming 8. p. 21 322 MAGNETISM AND ELECTRICITY LXVI mercurous sulphate and sulphuric acid, which will be washed away. The materials having been prepared, pure mercury and zinc amalgam are respectively poured into the two vertical parts of the H tube till the platinum wires are covered. The amalgam, which is solid at ordinary temperatures, should be heated till it is liquid, and the limb of the H tube intended' to contain it, heated to about the same temperature. It should then be poured into the H tube down a hot glass tube of outside diameter less than the inside diameter of the H tube, to prevent it soiling the sides of the H tube. The mercurous sulphate paste should then be forced down a glass tube on to the mercury, and the tube withdrawn, care being taken not to soil the H tube. A few crystals of zinc sulphate should then be placed on the surfaces of the paste and amalgam, and the rest of the tube up to about 1'5 cms. from the top filled with the concentrated zinc sulphate solution. A small quantity of clean paraffin wax should then be melted, and poured gently on to the surfaces of the solution in the tubes, till a layer about half a centimetre thick is formed. On the top of these layers, corks about '5 cm. thick should be placed, and the tops of the tubes then sealed with some hot resinous cement. A label bearing the date and the name of the maker should be attached to the stand on which the tube is sup- ported. The cell should stand a few days and then be compared with a standard cell, and the result recorded on the label. A disadvantage of the form of Clark cell which has been described, consists in the invariable cracking of the glass in the branch containing the zinc amalgam, at the point where the platinum is sealed into the glass. This is due to the expansion of the platinum as it slowly alloys with the mercury. In the cells recently constructed in the Physical Laboratory of the Owens College, the platinum wire has been covered electrolytically with nickel, to protect it against direct contact with the amalgam, and the results so far obtained seem to shew that this treatment gets over the difficulty. SECTION LXVIL COMPARISON OF ELECTROMOTIVE FORCES. Apparatus required : Two similar resistance boxes, high resistance, mirror galvanometer, Daniell, Leclanchd^ and Clark cells, and connecting wires. When the electromotive forces of different cells are to be compared together, it is necessary to carry out the comparison under similar conditions, and the condition usually adopted is that the cell under test shall be giving no current, or only an extremely small one, at the time of the test. The electromotive force E^ of the cell must therefore be balanced, the balance being indicated by a galvanometer in series with the cell remaining undeflected. The balancing electromotive force is best provided by the difference of potential between two points of a resistance through which a current is passing, and wires from the cell under test are brought to these two points. If it, is the resistance between the points of contact, C the current through that resistance, and E the difference of poten- tial between the points, we have El = CRi. If a second cell of electromotive force E^ is substituted, and balance exists when the resistance between the points of contact is /?,, we have, if the current is the same, E, = CR,. Hence E^ U^ which gives the ratio of the electromotive forces. 21—2 f 324 MAGNETISM AND ELECTRICITY LXVII Connect two similar resistance boxes A and B (Fig. 120), each of about 10,000 ohms, in series with a plug key K and two Leclanch^ ^ T I cells, taking out the plug from the ^11 '' key before making connexions. Set © up a Daniell cell, with a clean zinc j^" plate in zinc sulphate, and a clean copper plate in copper sulphate. To the terminals of one of the boxes, say j,. ^20. A J connect, in series with each other, the Daniell cell, a galvanometer G, a resistance R of about 100,000 ohms, and a spring key K.^, arranging that the cell under test, if alone, would send a current through the box A in the same direction as that sent by the Leclanch^s. Take out plugs for 10,000 ohms from the box A. Insert the plug in the key K, make connexion for an instant at the spring key Kc, in the test cell circuit, and observe the direction in which the spot of light moves. Take out plugs for 5,000 ohms from the box B, plugging 5,000 ohms in A so that the sum of the resistances of A and B remains the same. Make contact at the key K^, and notice the direction of motion of the spot. If it is the same as previously, take out more plugs from B and insert the same number in A. Continue adjusting the resistance of the two boxes, keeping the sum constant, till on closing the galvanometer circuit there is no deflection. If the arrangement is not sufficiently sensitive to enable the correct resistance to be found to within 1 ohm, determine it as nearly as possible, then short circuit the 100,000 ohms in the galvanometer circuit, and determine it more accurately. Make a note of the resistance in each box. Insert again the 100,000 ohms in the galvanometer circuit, then replace the Daniell by a Leclanch^ cell and balance as before. Then substitute the Clark cell, reading the tempera- ture of the air in the neighbourhood of the Clark cell, or better still, placing the cell in a water or oil bath, the temperature of which is measured. Since a Clark cell made according to the instructions contained in the Memorandum LXVII COMPARISON OF ELECTROMOTIVE FORCES 325 of the Board of Trade (Section LXVI.) has a difference of potential at its electrodes of 1-434 — '001 {t — 15°) volts, where t is the temperature centigrade, the electromotive force of each of the other cells can be found in volts. Calculate the electromotive force of the Daniell and of the Leclanche cells in volts. Arrange observations and results as follows : — 18 January, 1896. Resistance Boxes A and B. Cell Besistance A Resistance B Sum Daniell, No. 4 Leclauche, No. 13 Clark, No. 3, 19° C. Leclanche, No. 13 Daniell, No. 4 3,248 4,124 4,202 4,122 3,246 6,752 5,876 5,798 5,878 6,754 10,000 » » electromotive force of Clark at 19°= 1430 volt, electromotive force of Daniell 0947 4123 „ Leclanche = 1*43 x ^^^ = 140 The effect of temperature on the Daniell and Leclanche is masked by other irregularities, hence the temperature need not be noted and an accuracy of one per cent, in the result is sufficient. A compact fonn of the apparatus used in this exercise is known as a "Potentiometer." In it the re- sistance A consists of 15 equal resistances in i ± series, the one at the B end being a wire ~ along which the contact to the galvanometer slides. The contact from the cell to be tested is made by a switch moving over the 15 contact pegs at the ends of the 14 equal resistances. The arrangement is shewn in a diagiamniatic form in Fig. 121. The rough adjustment for a balance is made by means of the switch and the fine adjustment by means of the slider on the wire. ^ i Fig. 121. 326 MAGNETISM AND ELECTRICITY LXVII If the main current is supplied by a battery of two Leclanch^s, the above method may be used to measure any electromotive force less than about three volts, e.g.^ that at the terminals of a voltmeter intended for accumulator cells. When however the electromotive force to be measured exceeds a few volts, as, e.g., when a voltmeter reading to 100 volts is to be standardised, the arrangement requires modifying slightly because the electromotive force of the battery supplying the current cannot be raised above about three volts for fear of overheating the resistances A and B. The fall of potential between the terminals of the volt- meter is subdivided by connecting to the terminals a high resistance, divided into a number of parts the resistances of which bear known ratios to that of the whole. The difference of potential at the ends of one of these parts is then compared with that of the Clark cell by the above method. The circuit is arranged as shewn in Fig. 122, where V is the voltmeter 1-^0)—^ 1 Fig. 122. to be tested, R a variable resistance through which it is connected to a battery E, r the high resistance, r^ the portion of it down which the fall of potential is measured. If e is the value found by experiment, the electromotive force at the ter- minals of the voltmeter = — e. SECTION LXVIII. THE POTENTIOMETER METHOD OF MEASURING CURRENTS. Apparatus required : Standard low resistance, adjustable resistance, storage cells, current measuring instrument to he standardised, Clark cell and mirror galvanometer. When an electric current of A amperes is sent through a resistance of R ohms, it creates a difference of potential of V volts between the ends of the resistance, where V=AR, and sometimes it is more convenient to measure the current by means of the difference of potential it produces at the euds of a known resistance, than to measure it directly. This method is known as the " potentiometer method." In order that the method may give accurate results, the resistance R should be made of a material having a small tempemture coefficient, and should have a sufficiently large surface to prevent the tem- perature rising more than a few degrees. We shall shew how the method is used to standardise a current metre. Ck)nnect the current measuring instrument ^, to be tested, Fig. 128. to a standard low resistance R, capable of carrying the current which it is propo.sed to use without undue heating, and to an adjustable resistance /J,, using a number of storage cells E to supply the current required (Fig. 123). 328 MAGNETISM AND ELECTRICITY LXVIII Connect the terminals of a voltmeter which has been standardised (Section LXVII.) to the ends of the standard resistance. If such an instrument is not available, or if it is necessary to carry out the estimation of current to a high degree of accuracy, connect the ends of the resistance through a mirror galvanometer and tapping key to the ends of one of two similar resistance boxes A' , B' arranged in series, and forming a circuit with two Leclanche cells. Find, as described on p. 324, the plugs which must be taken out of the two boxes to enable contact at the key to be made, without a deflection of the galvanometer resulting. Now substitute in the galvanometer circuit a Clark cell e for low resistance R, and again balance by adjusting the resistances in the two boxes keeping their sum constant. If a reliable "Potentiometer" is available, it may be used instead of the two resistance boxes to compare the electromotive forces as explained on page 325. Observe the temperature of the Clark cell, calculate its electromotive force at that temperature. From the two obser- vations of resistances in the boxes A and B calculate the electromotive force E at the ends of the standard resistance. If the resistance of the standard is R ohms, the current through the resistance and current measuring instrument is E/R amperes, and this should be compared with the current as registered by the instrument. Record as follows : 16 January, 1897. Standardisation of Ammeter No. 4. Standard low resistance C = '20 ohms. Clark cell No. 4 at 18° = 1-431 volt. Resist- Resist- Reading Cor- Connexion to ance m A' ance in E.M.F. Current on Am- meter rection Clark cell 6560 4551 1-431 Ammeter 956 10155 •207 1-03 1-01 -h-02 )j 1940 ifec. 9161 •418 2-09 205 + •04: Clark cell 6559 4552 1-431 SECTION LXIX. THERMO-ELECTRIC CIRCUITS. Apparatus required : Watei- baths, thermo-circuits, four- way key and tnh'ror f/alvanomet€7\ If a circuit consists of wires of different materials, and if one of the junctions of two dissimilar wires is heated, an electric current flows through the circuit, and continues to flow so long as the difference of temperature between the heated junction and the rest of the circuit is maintained. This electric current is due to an electromotive force produced by the inequality of temperature, and it is found, for small differences of temperature between the two junctions, to be nearly proportional to the difference. For gi-eater differ- ences, if ti is the temperature of the hot junction, t^ that of the rest of the circuit, the electromotive force e in a circuit of two metals is given by the equation = ^(«.-o(?'-''-|^") where A and T are constants depending on the two materials of the circuit, T being a temperature known as their " neutral temperature." To verify the above statements the apparatus shewn (Fig. 124) is provided. It consists of two vessels containing water, in which are placed two test tubes containing the junction of the wires to be experimented on, and thermometers for indicating their temperatures. The rest of each tube is filled with clean sand 330 MAGNETISM AND ELECTRICITY LXIX or with petroleum, to improve the thermal connexion of the junctions and thermometers with the water. Fig. 124. The circuits to be tested consist of lengths of iron, nickel, and platinoid wire, to the ends of which lengths of copper wire are soldered, and brought to binding screws placed on the board through which the test tubes pass. The binding screws should be joined by copper connecting wires to a four-way key, so that each circuit may in turn be connected to a galvanometer of about 50 ohms resistance. With a galvanometer of this resistance, the effect of the different resistances of the circuits may be neglected, and the deflections taken as proportional to the electromotive forces acting in the various circuits. Fill the two vessels with water at the temperature of the room, and connect the thermo-circuits in turn through the four-way key to the galvanometer. Verify that there is no current in any of the circuits. Now raise the temperature of the vessel which has no binding screws over it, about 10°C. and keep it constant for 10 minutes. Then connect the galvanometer to each circuit in turn, and determine the deflection, noting the temperature before and after each observation. Raise the temperature 10° C. further and repeat. LXIX THERMO-ELECTRIC CIRCUITS 331 Continue till the hotter vessel reaches about 70" C, then cool it by adding cold water, taking observations in the same way during the process. Record as follows : 29 January, 1889. 'o 'i # Deflections Copper iron Copper nickel Copper platinoid 17-0 28-5 -1-5 + 3-3 + 2-6 170 39-5 -30 6-3 50 171 52-0 -4-7 11-4 9-2 171 63-0 -6-2 14-9 12-4 171 700 -7-2 18-0 14-3 171 57-0 -5-5 12-6 10-2 171 49-0 - 4-5 &c. 9-7 ifec. 7-8 Ac. Represent the observations for each circuit by a curve taking tj — U as abscissae and the deflections as ordinates. The curves will be found to be almost straight lines, and from this it is seen, that if we put the equation (p. 329) into the form A e = J5:o(f,-fo)(l-t^i + 0,where^o = -4^and6 = 2E, b must be small. Assuming that 6 = 0, determine the values of Eq in galvano- meter scale divisions per degree, which most correctly reproduce the curves found experimentally, and taking its value for the copper wire circuit to be unit}^ find its values for the other two circuits. If more accurate results are required, a low resistance galvanometer must be used, and the resistance of each circuit be made equal, or the total resistance of the galvanometer and each circuit be taken into account in comparing the electro- motive forces of the circuits. SECTION LXX. MEASUREMENT OF THE MECHANICAL EQUIVALENT OF HEAT BY THE ELECTRICAL METHOD. Apparatus required :, Covered calorimeter, thermometers, heating coil, standardised ammeter and voltmeter, storage cells, watch. When the whole of the work done on a body is converted into heat, the amount of work done bears a fixed ratio to the amount of heat produced, in whatever way the work is performed, and the work which has to be done to generate one gram-degree of heat is, we have seen (p. 142), known as the "mechanical equivalent " of heat. To determine this quantity, any convenient method of generating heat by performing work on the body may be adopted, and it is proposed in this section to do work by sending a current of electricity through an insulated wire immersed in water. If A is the current passing through the wire, and E the electromotive force at the ends of the wire, the rate at which work is done on the wire per second is EC watts, and if the current flows for t seconds, the total work done = ECt joules = ECt . 10"^ ergs. If the water rises in temperature degrees, and the water equivalent of the calorimeter thermo- meter and coi\ = w, the heat generated, supposing no heat is lost by radiation etc. = wO. If / joules are necessary to generate one gram-degree of heat, we have EAt = wdJ, from which J can be found. Weigh the calorimeter and stirrer provided. Nearly fill the LXX MECHANICAL EQUIVALENT OF HEAT. calorimeter with water and weigh again. Weigh also the platinoid resistance coil, and support it from the wooden lid of the calorimeter, taking care that it does not touch the sides. Place a thermometer graduated in tenths of degrees in the water. Connect the coil through a standardised ammeter and an adjustable resistance to sufficient storage cells to furnish the current required. Connect a standardised voltmeter of known resistance to the ends of the resistance coil. Make circuit and see that the instruments give proper indications and that the thermometer shews a gradual rise of temperature. Break the circuit, stir the water well, and after a few minutes take obser- vations of temperature every half minute as described on p. 129. At the end of the first period put on the current, read the thermometer every half minute, observing the voltmeter 15 seconds before each minute and the ammeter 15 seconds after each minute. This second period should continue till the tem- perature of the calorimeter has risen about three degrees, then at the end of one of the intervals, the current should be switched off and observation of temperature continued till the rate of change is steady. From the first and third periods the cooling corrections during the second and third periods should be calculated as in pp. 129 — 130, and from the initial and final corrected temperatures the rise of temperature determined. J is calculated by substituting in the above equation for E the mean electromotive force, and for A the mean current, if both quantities show only small variations during the ex- periment. If they are not sufficiently constant, the product of EA must be calculated for each interval of time and the mean product substituted in the equation. . Record as follows: 10 December, 1897. Weight of calorimeter = 550 grams „ „ and water ... =245*2 „ water =1902 „ platinoid coil ... ... = 28 „ Water equivalent of calorimotcr find coil . = 7-8 334 MAGNETISM AND ELECTRICITY LXX Total water equivalent Rise of temperature . . . .*. Heat generated ... Mean electromotive force Resistance of voltmeter Mean current through ammeter „ „ „ voltmeter coil ... 1-45 14 = 198 grams = 311° C. = 616 gram-degrees = 1'45 volts = 14 ohms = 11 10 amperes = 10 „ ... = 1100 Time ... ... ... ... ... =161 seconds Work done = 1'45 x 11 x 161 ... = 2575 joules . •. Equivalent = ^-^ = 418 joules per gram-degree = 41'8 X 10^ ergs per gram-degree This value happens to be almost exactly right, but errors of one per cent, are likely to occur, unless the voltmeter and ammeter have been very carefully standardised. SECTION LXXI. INDUCTION OF ELECTRIC CURRENTS. Apparatus required : Two solenoids, one sliding within the other y tangent galvanometer, reversing key, low resistance mirror galvanometer, resistance coils. When a current is made, broken, or altered in strength in any circuit, induced currents are produced in neighbouring circuits, and it is the object of this exercise to find on what conditions the magnitudes of these induced currents depend. The induced currents will last only a very short time, and a galvanometer in one of these neighbouring circuits will not indicate the strength of the current, but the total quantity of electricity which has passed through it. If a is the angle of the first swing of the galvanometer needle produced, it can be shewn that the quantity of electricity which has passed through the coil of the galvanometer is proportional to sin ^ . If the galvanometer needle hangs in its proper position when no current passes through the instrument, and the scale is properly adjusted, where x is the observed deflection and d the distance of the scale from the mirror (page 160). If an extreme error of a half per cent, is considered allowable, and the deflections ob- served on a scale half a metre in length, placed a metre from the mirror, never exceed 25 cms., the second term on the right- hand side is negligible ; and we may therefore take the observed 336 MAGNETISM AND ELECTRICITY LXXI reading x to be proportional to the quantity of electricity which has passed through the galvanometer. The experiments of the present exercise are supposed to be made under these conditions. Two solenoids, P, S (Fig. 125), mounted on blocks of wood so that their axes are coincident, are provided. The inner coil Fig. 125. P can be moved in a direction parallel to the axis, and placed with its centre at any convenient distance from the centre of the outer fixed solenoid >S^. Each solenoid is divided into three parts, and the number of turns to each part should be counted and recorded. Arrange the turns of the inner coil in series with each other and with a cell, a reversing key, an adjustable resistance, and a tangent galvanometer. Place it within the outer coil so that the centres of the coils coincide, the mark on its base will then read on the scale of the outer coil. Connect, by means of fairly stout copper wire, the end terminals of the outer coil through a resistance box to a low resistance mirror galvanometer. At first cut the mirror galvanometer out of circuit by connecting the two wires leading to it to the same terminal, and observe whether making or breaking the battery circuit has any effect on the galvanometer. If so remove the coils further away and put their axis in such a position that this is no longer the case. Adjust the resistance in series with the inner coil till the current flowing through it is one ampere, and take readings occasionally to see that it remains steady. Notice that making or breaking the battery circuit produces a deflection of the mirror galvanometer needle. Observe the extent of the first swing and LXXI INDUCTION OF ELECTRIC CURRENTS 337 verify that the swing on making is equal and opposite to that on breaking the primary, t.e. the battery circuit, and that the swing on reversing the primary current is double the previous swings. If this is found not to be the case, cut the galva- nometer out of circuit as previously and make sure that there is no direct action from the primary coil. Effect of the relative positions of the two coils. Arrange the resistances so that on making or breaking one ampere in the primary, the deflection is about half the greatest observable deflection. Determine its amount, then slide the inner coil through 1 cm. in a direction parallel to the axis of the coils, and again determine the deflection. Repeat for 2, 5, 10, 15 and 20 cms., thus gradually sliding the inner coil out of the outer one. Now rotate the movable coil till its axis is bisected at right angles by that of the fixed coil, and verify that there is no deflection on making or breaking the primary circuit. Move the inner coil towards the outer, keeping it parallel to itself, and verify that by proper adjustment a direction of the axis may be found, for which there is no doflection, even when the coils are near together. Effect of the resistance of the secondary circuit. Place the movable coil within the fixed coil in the position in which the induced current has been found to be a maximum, and deter- mine the swing on making or breaking the primary circuit. Double the total resistance in the mirror galvanometer (or secondary) circuit by taking out plugs from the box equal to the sum of the resistances of the galvanometer, coil and con- necting wires, and verify that the swing is now half what it was before. Effect of the magnitude of tfie current in the pj-iniary. Increase the resistance in the battery circuit till the current has half its original value, and verify that on making or breaking, the swing is half what it previously was. Effect of number of turns in the primary. Adjust the resist- ance till the primary current is again one ampere. Change the connexions of the primary so that the current passes through tiitro coils only. The connexions are arranged so that the s. p. 22 338 MAGNETISM AND ELECTRICITY LXXI resistance in cii-cuit, and therefore the current, remains the same; but if any variation is observed, adjust the resistance in series with the storage cell till the current is again one ampere, and observe the swing on making or breaking. Notice that it is decreased in the ratio 3 : 2. Now connect to the two centre screws so as to send the current through the central coil only. Notice whether the current remains the same ; if so the deflections will be found to be again decreased in the ratio 2:1. Effect of number of turns in the secondary. Returning to the whole of the turns on the primary, change the connexion of the galvanometer from 3 to 2 coils of the secondary. The terminals of the coils are so arranged that the resistance in circuit remains the same. Verify that on now making or breaking the primary circuit, the swing is decreased in the ratio 3 : 2. Make connexions to the centre coil only, the ar- rangement being again so that the resistance remains constant. The deflection will be found to be decreased in the ratio 2 : 1. It has therefore been shewn that for a given position of the coils with respect to each other, the induced current in the secondary is proportional to the current in the primary, to the number of turns of the primary, to the number of turns of the secondary, and inversely proportional to the resistance of the secondary circuit. Effect of moving magnets. Instructive experiments maybe made, when two observers are available, by moving magnets in the neighbourhood of the secondary coil. For this purpose the coil should be removed to a distance, such that the bar magnet, to be used in the experiments, has no direct effect on the galvanometer. Place the magnet in the centre of the coil, let one observer remove it quickly to a distance from the coil, and let the other observer note the deflection produced. Replace the magnet in its original position and withdraw it again, but pulling it out towards the other side. This should produce an equal deflection and in the same direction. Place the magnet outside the coil and in such a position that its axis coincides with that of the coil ; note the deflection when it is suddenly moved to a distance, and shew that it is LXXI INDUCTION OF ELECTRIC CURRENTS 339 the same as if the magnet were turned through a right angle about its centre. Verify that when a magnet is moved quickly from a position A to a. position B the effect on the galvanometer is the same» through whatever path the magnet is moved. Also try to shew that when the magnet can be moved from a position A along any path back to the same position -A in a time which is short, compared to the time of oscillation of the galvanometer needle, the total effect on the latter is nil. 22—2 SECTION LXXIL DETERMINATION OF THE INDUCTANCE OF A COIL. Apparatus required : Coil of considerable self-induction. Post Office resistance box, additional resistance, mirror galvano- meter, voltaic cell of constant electromotive force. If one of the arms of a Wheatstone bridge possesses self- induction, the galvanometer will, even when the bridge is balanced for steady currents, shew a temporary deflection, when the strength of the current through is altered. If L is the inductance of the coil and x the rate of change of the current passing through it, the effect on the galvanometer is equal to that which would be produced if an opposing electro- motive force Lx were introduced into the arm in which the coil is placed. Such an electromotive force would cause a current KLx to flow through the galvanometer, where the value K depends on the resistances of the galvanometer, and of the arms of the bridge. If the current is suddenly changed from to X, the electromotive force at each instant depends on the rate of change, but the total quantity of electricity, which flows through the galvanometer, will be KLx, which is the same as that conveyed by a constant current KLx, flowing during a short time t, such that xt = x. This quantity of electricity passing through the galvano- meter coils, will cause a deflection or " throw " of the galvano- meter needle, the extent of which may be calculated. If the moment of inertia of the suspended system of magnets be /, and (o denote the initial velocity, ^Ico^ is the initial kinetic energy. The diminution of kinetic energy during the swing must be equal to the work done against the magnetic LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 341 forces tending to bring the system back into its position of equilibrium. To find an expression for that work, we may think of the analogous case of a compound pendulum. If the weight of such a pendulum be W, and the difiference in level of the centre of gravity between the displaced and equilibrium position 6, the work done will be Wb, which is equal to Wh ( I — cos 6), where 6 is the angular displacement, and h the distance of the centre of gravity from the point of suspension. Hence for the greatest angular displacement (a), for which the kinetic energy is zero, the work done must equal the original kinetic energy, or /&)- = 2ir/t (1 - cos a) = 4Tr^sin» |, .-. a, = 2y -p.sin^. Returning to the case of the oscillating system of magnets, we need only replace Wh by the moment of the couple, tending to bring the magnets back, when the displacement is a right angle. Representing this quantity (which is equal to the product of the earth's horizontal force and the magnetic moment of the system) by H, we obtain for the relation be- tween the initial angular velocity and the greatest angle of displacement, a .-V? sin^ If G denote the couple acting on the galvanometer needle in its position of rest, when unit current pivsses through its coils, a current i will cause an angular acceleration Cri/I, and if the duration of this current be t, which is supposed to be small compared to the time of oscillation T of the galvanometer needle, the angular velocity generated will be Oir/I, where we may replace it by the total quantity of electricity Q, which has passed through the galvanometer coils. Substituting this angular velocity in the above equations we find 342 MAGNETISM AND ELECTRICITY LXXII The moment of inertia / may be determined or eliminated, by observing T, for Hence The value of HIG can be determined experimentally by observing the permanent deflection 6 of the galvanometer, produced by a known current z, for in that case (p. 243) Gi^Htaxie, provided that the galvanometer needle moves in a field which is sufficiently uniform for the tangent law to be true (see page 244). Otherwise tan 6 must be replaced by some function of 6, obtained by calibrating the instrument. If the simpler supposition is sufficiently accurate, IT tan 6 We have deduced the value of a under the condition that the needle moves without damping, which is never quite the case. But the damping may be taken into account by multi- plying the observed amplitude by e^'"^, since the amplitude of an oscillating system is reduced in the time t in the ratio of e-^tiT.i^ p. 282, and the time taken up by the needle, in moving from its position of rest to its first greatest elongation, isr/4. The final equation for Q is therefore ^zV/2siu| TT tan 6 To apply this equation to the determination of the inductance of a coil we substitute for Q the value KLx, which has been shewn to be equal to the quantity of electricity flowing through the galvanometer, when the current in the branch of the Wheat- stone bridge containing the self-induction L is suddenly changed LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 343 from to X. The quantities K and % are eliminated in the following way. If the balance of the bridge be disturbed, by inserting a small resistance r in series with the coil, this addi- tional resistance will, while a current x' is passing through it, have the same eftect on the galvanometer as an opposing electromotive force rx. If this electromotive force produces a current i in the galvanometer, then i = Krx'. Hence L = r — , ^ . X TT tan V This equation holds within the limits of deflection for which the tangent law is true. In many cases the calculation may be further simplified. The ratio x'lx is the ratio of the currents in the coil with and without the additional resistance r ; the latter quantity should always be so small that x is sensibly equal to x. If a and Q are sufficiently small these angles may be taken to be proportional to the observed deflections d^ and 2), otherwise sin ^ and tan 6 must be evaluated according to the equations given on p. 160. But it follows from these equations that the ratio sin a/2 / tan B is equal to c?i/2(/o to the second approxi- mation if ■|V¥=— This may be secured by a proper choice of r. The coefficient e^^^* may be expressed in the form of a series, the first three terms of which are ^ + 2+8- The third term is often negligible, e.g. when an accuracy of !•/, is considered sufficient and X is smaller than '2, or when an accuracy of P/o is required and \ is smaller than 06. The determination of X is best carried out by a separate series of observations, but when that quantity is small, and 344 MAGNETISM AND ELECTRICITY LXXII when the experiment is made rather as an exercise than for the purpose of obtaining the best possible results, time may be saved by observing not only the first throw (d^), on making or breaking circuit, but also the second deflection (d.^) on the same side. The equations on p. 282 give for this case d^/d^ = 6^^ and hence e^/*^ may be replaced by 1--^] . When di and d^ are nearly equal we may simplify further, for r =(■*''■**)' - 1 4. ^ d,-d, 5^ ( d, - cg, y ~ "^4 d, "^32 1 d, ; "^••• If the third term of the series can be neglected, we obtain d,e'l^ = d,-\-l{d,-d,). If an accuracy of one per cent, is required, this simplification should not be introduced, unless d^jd^ is smaller than \'2 which corresponds approximately to X smaller than "1. Taking account of these simplifications we shall be able to use one of the following three equations whenever X is smaller than '2, i.e. d^jd^ smaller than 1*5, and d^jd^ is approxi- mately equal to 1"17. r_Trd, (d,\i 27rI»'W ' ^-^- D • The first equation is the one to apply, when accuracy is of greater importance than time, so that an independent de- termination of X may be made. The second is used when X is not separately determined, but when the second deflection is observed ; while the third applies to the same case with the additional proviso, that djdz is smaller than 1*2. Care should be taken, to have the zero reading of the galvanometer near the centre of the scale and to take the LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 345 deflections towards both sides, thus eliminating any errors due to faulty adjustment of the scale. If X is to be determined by experiment, we make use of the equation (p. 283) log s, - log Sn = (n-1)\ log e, 8i and Sn being the first and the nth arc of swing. It is found that an error of reading has the smallest effect when 5,/5„ = e=2-7. This suggests the following procedure. Set the galvano- meter needle into vibration, and read the first deflection (i, on the right and the first deflection rf/ on the left. Then wait until the deflection has diminished to about ^ and read again deflections dn and rf,/. Assuming as a first and generally suffi- cient approximation, di -|- d^ and rf„ + dn, to be proportional to the arcs of swing, we may, knowing n, apply the above equation to the calculation of \. The result may be rendered more accurate by observing and combining in a similar manner 5, and «„+!, «s and s^+j etc. If the oscillations are too rapid to allow the deflections toward both sides to be accurately ob- served, we must divide the experiment into two parts, observing first deflections to the right and applying the equation log rfi — log rf„ = 2 (/I — 1 ) \ log e to deduce the value of \. Observing similarly the deflections to left we obtain a second value for X, the mean of the two being taken for the final result. When \ is so sfnall, that it would take too long to wait till the amplitudes are reduced in the above ratio, we may procecnl as in the measurement of time intervals, observing say six deflections, waiting until six oscilla- tions have passed unob.served, and observing six further ones, which are combined with the six first in the manner already explained. The principal experimental difficulty will be found to be due to the gradual heating of the coil by the current, which must be sufficiently strong to give a measurable throw on 346 MAGNETISM AND ELECTRICITV LXXII making or breaking the circuit. Subject to this condition, the current through the coil should be as weak as possible. It is important therefore that the arms of the bridge should be arranged, so that the galvanometer is most sensitive to variations of resistance in one arm of the bridge, for a given current in that arm. This condition leads to results, which are quite different from those which apply when the electromotive force is given (p. 287). The best arrangement of the bridge, when S is the resistance whose inductance is to be determined, is that in which the resistance we have denoted by Q in the previous exercises is as small as possible, and that denoted by R as large as possible \ One of the galvanometer terminals should be connected to the junction of P and Q, the other to the junction of R and 8. The best resistance of the galvano- meter of a given type is G^ = p+e or when P is large compared to Q, G = S -\- Q, S being the resistance of the coil. The sensitiveness of the arrangement for given values of P, Q and G, as compared to the best possible arrangement, is given by the fraction where k is the ratio of the actual to the best possible galvano- meter resistance. A discussion of this equation shews that the sensibility is not much reduced, if k is intermediate between 4 and i', also that it will be sufficient to make the ratio PIQ== 10. But care should be taken that Q never exceeds S and it would be better if S were at least three times as great as Q. The change of resistance r may be obtained by connecting a high resistance in parallel with the coil, when the steady deflection is taken. If W be the resistance of the shunt, r is equal to S'''/{S -}- W). The shunt must be disconnected 1 Arthur Schuster, "Electrical Notes," Phil. Mag. p. 175 (1894). LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 347 when the throw is taken. In order to avoid a change of connexions during the observations we may vary the resistance of R instead of that of S, remembering that a change r in the resistance of the coil is equivalent to a change PrjQ in the resistance R. If r' is the change in R, which produces a steady deflection 0, we must therefore substitute for r the quantity r'Q/P. To carry out the experiment, arrange the given coil S to form the fourth arm of the Post Office bridge; Q being the ratio arm adjoining S. Make Q=10 and P=1000. Connect the galvanometer through a commutator to the junction be- tween P and Q, and to that between R and S. Introduce also a commutator into the battery circuit, so that the current may easily be reversed. It is important that the battery should be one giving a constant electromotive force, e.g. a Daniell or storage cell. Leclanche cells are too variable. Adjust the zero of the galvanometer to the centre of the scale nearly. Leave the galvanometer circuit open, pass the current through the coil and notice whether the galvanometer shews a deflection. If it does, the coil acts directly on the needle and its position should be changed so that the action ceases. This is most easily secured by placing the coil with its plane nearly horizontal and tilting it till the magnetic field due to it is vertical at the galvanometer. Now close the galvanometer circuit and balance for steady currents. When the balance is approximately made, break the battery circuit and take an observation of the throw, which should be between 10 and 20 cms. as measured on the scale. If the throw is too large, the electromotive force of the battery should be diminished or a resistance inserted in the battery bninch. If it is too small, the galvanometer should be made more .«iensitive, and if that is not possible the electromotive force of the battery must be increased. Having obtained an approximate value of the throw, the change of resistance r' in the arm iJ, which gives a steady deflection of between I'l and 1*2 times that of the throw, should be found. If the galvanometer has a small logarithmic decrement and takes a long time to come to rest a small coil in series with an 348 MAGNETISM AND ELECTRICITY LXXII auxiliary battery and key may be fixed behind it, and con- nexion may be made and broken in such a way as to bring the suspended system to rest. The above preliminary experiments having been concluded, note the position of rest of the galvanometer needle. Obtain as accurate a balance of the bridge as possible. If there is a slow creeping of the spot of light, while the steady current passes through the coil, it is probably due to the heating action of the current, which slowly increases its resistance. If that is the case, adjust the balance so that the spot of light is deflected a little to the side opposite to that towards which the creeping takes place. Wait till owing to the heating action the spot of light occupies accurately the zero position; then suddenly break the battery circuit and observe the throw, reading both the first deflection (c^) and that which follows it on the same side (d^). Make the battery circuit again, change the resistance R by the addition of r' and observe the steady deflection. Break the galvanometer and battery circuits and read the zero again. Reverse the commutator in the battery circuit and repeat the observations, both for the throw and the steady deflection. Reverse the galvanometer current and again repeat the observations, first with the battery commutator in one and then in the other position. Four sets of readings are thus obtained the means of which must be combined in the final result. Determine the time of oscillation of the galvanometer needle by observing the time of say 50 swings if the time be short and the logarithmic decrement small ; or if the time be long, observe the instant of passage through zero for say six swings, allowing an interval of six, then observe six more and combine in the usual way. If it is desired to determine the logarithmic decrement this should now be done by one of the methods explained above, care being taken that the sensitiveness of the galvanometer is the same as that used during the experiment. The logarithmic decrement for the same galvanoineter is proportional to its time of oscillation. LXXII DETERMINATION OF THE INDUCTANCE OF A COIL 349 Tabulate your results as follows : 2 January, 1898. Inductance of coil A. Galvanometer No. 327. P = 1000. Q = 10. 2 Daniell Cells. Balance obtained with i2 = 2194. On breaking circuit First swing to right (mean of four observations) = 11*04 cms. Second „ „ „ „ „ = 7*56 „ Changed R to 2255 ; r = 61, .-.?'= 61. Steady Deflection (mean of four observations) =12*11 „ Time of oscillation = 3*44 sees. J _ n. 3*44 1104 /ll*04\i • ^-'^^ 27r-i2lll75¥; = 335 henries. SECTION LXXIII. LEAKAGE AND ABSORPTION IN CONDENSERS. Apparatus required : Condensers, one of which is specially selected to shew leakage and absorption, high resistance galva- nometer, discharge key, cells. Unless the plates and terminals of a condenser are well insulated, the condenser is found gradually to lose its charge. The rate at which this loss takes place is generally greater, the greater the charge which the condenser possesses. To determine the extent to which leakage takes place in a condenser C (Fig. ]27), connect it through a " condenser key" K to one or two Leclanche cells L and a galvano- meter G, as shewn in the figure. <«)• HI When the movable part of the key I iL is down, the condenser is in connexion ^ig- 127. with the cells and is charging; when the disc marked " insulate "^ on the key is pressed, the terminals of the condenser are in- sulated from each other ; when the disc marked " discharge " is pressed, the terminals are connected together through the galvanometer, and the condenser discharges itself. Owing to the passage of the discharge through the galva- nometer, the needle is deflected, and the extent of the first swing from the position of rest is proportional, within the degree of accuracy required for the present purpose, to the amount of electricity discharged through the galvanometer. Take several observations of the swings after the operation of charging has been performed in the least possible time. In order that the condenser may be properly discharged between each observation, allow the key to remain at the discharge LXXIII LEAKAGE AND ABSORPTION IN CONDENSERS 351 position for three minutes before again charging. The mean of these deflections may be taken to represent the charge of the condenser due to the applied electromotive force before leakage has had time to occur. They ought not to be less than 20 large scale-divisions. Now take several observations, charging the condenser for an instant only as before, but insulating for ten seconds, then discharging, and allowing three minutes before again charging. Repeat the observations for instantaneous charges followed by intervals of 30 seconds, 1, 2, 3, 4, and 5 minutes' insulation, and plot the results in the form of a curve, taking times as abscissae and de- flections as ordinates. Take logarithms of the de- pjg 128. flections, subtract each from the logarithm of the first deflection, divide by the differences in the time of insulation, and tabulate the results as follows : Condenser A. Leakage. 30,. tJy^^J r...^.. ■-•n ;"■ 20 I .rge I 10 %N i Ni^ ^-^ — Charged "Smts. 1 ^^ 1 1 0^ 1 2 . . .X ' Time of Deflection Log. of Difference 1 insolfttion Osec. on discharge deflection Difference Time 30-9 1-490 10 „ 23-3 1-367 •123 •0123 30 .. 15-8 1199 •168 •0084 1 min. 90 •954 •245 •0082 »7 5-5 •740 •214 •0036 3 „ 3-6 •556 •184 •0031 * » 2-9 •470 •086 •0014 6 ,. 2-4 •380 •090 O015 If the leakage at any instant were proportional to the charge in the condenser at that instant, the numbers in the last column would be equal. Their gradual diminution shews that the leakage increases more rapidly than the charge. In order to test whether the inequality is due to absorption of the charge by the dielectric of the condenser, the effect of 352 MAGNETISM AND ELECTRICITY LXXIII variation in the time of charge should now be found. The condenser should be charged each time for 10 seconds instead of an instant only, and the rest of the experiment carried out as previously. Then the charging should be continued for 30 seconds and afterwards for 1 and for 5 minutes. Tables of results for each time of charge should be given and leakage curves drawn alongside the one drawn previously. If absorption by the dielectric occurs, the absorbed charge should make its reappearance after discharge on the insulated conductors of the condenser. To test this, if the condenser has a short circuiting key, insert it fur an instant to produce the first discharge without affecting the galvanometer, or if there is no key, arrange one in parallel with the condenser. Charge the condenser for 10 seconds, discharge for an instant through the short circuiting key and then insulate for 5 seconds. On now pressing the discharging key, the galvanometer will be deflected owing to the passage through it of the ''residual" discharge. Repeat the observations, allowing the condenser to remain insulated 10, 20, 40, and 60 seconds before taking the residual discharge. To find the influence of the time of charging on the magnitudes of the residual discharges, charge the condenser for 10, 20, 40, and 60 seconds, insulate in each case for 5, 10, 20, 40, 60 seconds and take the first residual discharges as before. Express the results by curves and in tabular form, as follows : Condenser A. First Residuals after charge for given time, discharge, and subsequent insulation for given time. Time of charge First Residuals after insulation for 5 sees. 10 sees. 20 sees. 40 sees. 60 sees. 5 seconds 10 „ 20 „ 40 „ 60 „ 8-0 10 11 13 14 9-0 11 13 16 18 10 12 14 17 20 11 14 17 20 23 12 17 20 23 26 LXXIII LEAKAGE AND ABSORPTION IN CONDENSERS 353 ! ' 1 30 1,11 RMidiMljChwfM ; I 20 . 210.8ms. 1 ^ r^ ^ 10 / r^ _u —> ss. 1 C^ ' ■ 1 ' ! 1 10 20 Peri 30 l40 ISO |60^ecofKt» >do/in.^lat•op. 1 1 Fig. 129. To test for successive residuals, charge again for 15 seconds, then discharge through the short circuiting key. Insulate for 15 seconds, take the first residual, insulate for 15 seconds more and take the second residual, and so on till no more charge remains in the condenser. Tabulate as follows and draw a curve shewing the magni- tudes of the successive residuals as ordinates, and taking times of insulation since the first discharge, as abscissae. After insolation for Residual discharges 15 seconds 15 sees, more 15 „ „ 15 ., „ 7-2 6-4 5-2 41 Take a second condenser not specially chosen to exhibit leakage and absorption, and test whether the leakage and absorption are small. 8. p. SS SECTION LXXIV. COMPARISON AND USE OF CONDENSERS. Apparatus required : Condensers, discharge key, high resistance mirrcyr galvanometer, cells. It has been stated in the previous section that when a condenser is discharged through a galvanometer, the extent of the first swing from rest is proportional to the quantity of electricity discharged. If Q is this quantity, V the potential of the battery used to charge the condenser, and C the capacity of the condenser, then Q = CV. 'J'he method of discharge may therefore be used either to compare the capacities of a number of condensers charged by the same battery and discharged through the galvanometer, or to compare the electromotive forces of a number of cells used to charge the same condenser. Apply the method, using a Leclanche cell and a high resist- ance galvanometer which swings without too much damping, to determine the capacities of the condensers provided, assuming that of the standard condenser to be given in microfarads. The condensers should be placed in circuit in turn, charged for an instant, then discharged, and the extent of the swing of the galvanometer needle observed. In order to see whether leakage has any effect on the observations, take discharges from each condenser after 10 and 20 seconds' insulation, as well as instantaneous discharges. LXXIV COMPARISON AND USE OF CONDENSERS 355 Record as follows: 19 January, 1897. Nalder Galvanometer No. 5232. Leclanch^ Cell No. 12. Condenser Standard Swiss Paraffin No. I. II. III. Instan- Capacities taneous 10 sees. 20 8608. micro- farads 23-7 23-8 23-7 -33 26-4 25-2 250 •37 27-2 261 25-4 -38 Remarks no leak small leak Select the condenser which shews least leakage, charge it in turn by means of a Leclanch^, a Daniell, a Clark and a storage cell. Take two discharges through the galvanometer with each cell, and assuming the electromotive force of a Clark to be 1'434 volts, calculate those of the other cells. Record as follows : Swiss Condenser. Nalder Galvanometer No. 5232. CeU Deflections Mean E.M.F. Leclanch6 Daniell Storage Cell Clark 34-9 35-0 24-2 241 48-1 47-5 351 35-2 34-95 2415 47-80 3515 1-42 voltiL •98 „ 200 „ 1-434 „ 23—2 SECTION LXXV. DETERMINATION OF THE CAPACITY OF A CONDENSER. Apparatus required : Condenser, high resistance galvano- meter, resistance coils, voltaic cell. When Q coulombs of electricity are discharged through a galvanometer the needle is deflected and swings through an angle a which, neglecting the damping, has been proved (page 342) to be given by the equation where HjG is the constant of the galvanometer. If the quantity Q is the charge of a condenser of capacity (7, due to an applied electromotive force E, then Q = GE and therefore ^ I H T . a ^ = E'G'^''''2' The damping may be taken into account by introducing the factor 1 TT e^l'^ on the right-hand side. The quantity -^ . -^ may be deter- mined or eliminated, by observing the steady deflection of the galvanometer produced by the electromotive force E, acting through known resistances. Let a circuit be arranged as in Fig. 130, the cell being the same as that which has served to charge the condenser, and having its terminals connected to a resistance jR, which should be large compared to that of the cell, so that the difference LXXV DETERMINATION OF THE CAPACITY OF A CONDENSER 357 of potential at the ends of R, may be taken to be equal to the electromotive force E. The terminals of the galvanometer are connected only to a small portion r of R, so that the difference of potential at the ends of the galvanometer circuit is ErjR. If the resistance of the galvanometer be p and the resistance p shewn in the figure be ^*^- ^^• not inserted, the current passing through the galvanometer will be Er/pR, and the permanent deflection will be given by the equation •'• GE~ pRtaind'' rTe^sinl irpR tan 6 ' The galvanometer resistance p is probably known from previous determinations, but may, if desired, be obtained ap- proximately or eliminated in the above expression, if a known resistance p' is inserted in the galvanometer circuit and the steady deflection d' observed. We have in that case Er H ^ ^ = ^ tan ^ ; (p-¥p)R G and combining this with the equation obtained for the case when p' is not inserted, we find p tan d' ^""tan^-tan^* The condition most favourable for the determination of p by this method, is that in which p and p are equal so that the introduction of p approximately halves the deflection. The remarks made in Section LXXIL, page 34:^ as to 6*''8m rt the best way of determining a apply to the present 358 MAGNETISM AND ELECTRICITY LXXV section, and to vary the experiment the quantity X should in the present exercise be independently determined. Determine the capacity of the condenser provided, charging it by two Leclanche cells and discharging it through the galva- nometer. Take four observations of the first throw, two of which should be made with the battery reversed. Connect the galvanometer to the two Leclanche cells as shewn in Fig. 130, adding a commutator in the battery branch. Make R about 10,000 ohms, and r such that a deflection of about 1*2 times the discharge throw is obtained when p=0. Measure the deflection with the battery current in both direc- tions. Insert a resistance p' such that the deflection is about half what it was in the previous case, and again make two determinations of the deflection, reversing the battery current between the two observations. Measure the time of swing and the logarithmic decrement, following the instructions given on page 345. Record your observations as follows : 3 October, 1900. Standard Condenser marked 1/3 microfarad. Galvanometer No. 5232. 1. Discharge of Condenser after being charged by two Leclanche Cells. First deflection (mean of two deflections in each direction) = 165 cms. 2. Measurement of steady Deflection. iR = 10000, r = 40. (a) p' = 0. Deflection (mean of one obser- vation in each direction) = 18*35 „ (6) p' = 10000. Deflection (do.) = 7-45 „ Hence galvanometer resistance 7-4.^ ^ = ^"""" 18-35- 745 -^««Q- A previous direct measurement of the galvanometer resist- ance had given p = 6570. The latter value, being more accurate, is adopted in the calculation. LXXV DETERMINATION OF THE CAPACITY OF A CONDENSER 359 Time of 20 swings (a^ ^erage of 3 observati ions) = 749 seconds „ one swing = 3-745 ,. 4. Determination of logarithi mic decrement. Successivt ; deflections to right to left (1) 1G16 19-58 (2) 10-33 12-37 (3) 680 8-34 (4) 4-61 5-41 each number being the mean of three observations. Ratioof(3):(l) 1 = 2-38 and 235 „ ., (4) : (2] 1 = 2-24 „ 2-29 Mean Ratio 2-31 ^_1 log 2-31 _ i X -3636 X 2-302 = •2092 4 loge -t = 1-105 40x3-745 X 1105 x 1615 10000 X 27r X 6880 x 1835 = -337 X 10-« = *337 microfarad. APPENDIX. The following details, referring chiefly to the dimensions of the apparatus used in Owens College, will probably prove useful to teachers. SECTION IV. V. VI-VIII. XI. XI 11. Spirit Level. The ordinary 8 inch " brass adjusting level " is suitable. If the tube is not graduated, a paper scale may be gummed to it. Calibration Tube. A tube about 20 cms. long, of '7 cm. external and '07 cm. internal diameter, may be used. The tube and mercury should be quite dry. Balance and Density. A piece of quartz of 100-200 grams is a suitable body to weigh, since it can easily be kept clean, and therefore of constant weight. As quartz is unacted on by water it is also suitable for the density determinations. Moment of Inertia. The block should be suspended by means of a thin soft wire free from kinks. Compound Pendulum. A brass bar about a metre long, 2 cms. broad and -5 cm. thick, is suitable. The steel knife- edges are supported on glass plates. APPENDIX 361 SBCnON XIV. XV. XVI. XVII. XVIII. XIX. XX. Beams. Straight-grained beams about a metre long, 1 -3 to 2 cms. broad and -7 to -9 cm. thick, should be used. Rigidity. The thinnest steel pianoforte wire (No. 30) and a wire about No. 25 are suitable. The ends of the wires are soldered to short lengths of brass wire of 1*5 to 2 mms. diameter, one of which is tapped and screws into a hole in the axis of a cylindrical brass weight about 4 cms. diameter and 5 cms. long, and the other is clamped in the support. Viscosity. The tubes should have a length of 50 to 80 cms., the bore for water should be '5 to "8 mm., and for the calcium chloride solution 1*2 to 1*8 mms. A solution of density 11 5 to 1-20 (about 20 Vo) »» suitable. Surface Tension. The thin glass used tor covering microscope slides is suitable for the balance method. Expansion of a Solid. Tubes of about 1 cm. diameter and 60 or 70 eras, long are convenient. Expansion of a Liquid. The graduated stem of a broken 0° to 100* C. mercury thermometer serves for the sttMu of the dilatoraeter. The bulb should have about 4 times the volume the bulb of the thermometer hod. Pressure Coefficient op a Gas. A little strong sulphuric acid should be placed in the bulb to keep the air dry. By warming the bulb slightly after a little mercury has been poured into the open tube, the volume of air enclosed may be reduced to the desired value. 362 APPENDIX SECTION XXI. XXII. XXIV. XXV. XXVI. XXVIII. XXIX. XXX. Expansion op a Gas. A tube of -l cm. bore and about 25 cms. long is suitable. The divisions etched on it may lt>e ren- dered more distinct by rubbing a little rouge into them. A strip of white enamelled glass should be placed between the tube and frame. Pressure and Boiling Point. A wide-necked pint bolt-head should be used for the boiling flask. A condenser 40 cms. long is necessary. Laws of Cooling. The calorimeter of copper -3 mm. thick is 6 5 cms. diameter and 8 cms. high. It stands on cork legs within a water-jacket, the inside diameter of which is 8*5, the outside 12*5, and the height 12 cms. Calorimetric Measurements. A piece of rubber 1x3x3 cms, is convenient. Specific Heat of Quartz. The steam heater is 40 cms. long, has an in- ternal diameter of 6*5 and an external of 9-5 cms. The internal cavity is closed at the top by a cork, and at the bottom by a metal door which can be moved aside so that the substance may be lowered into the calorimeter. Latent Heat op Steam. The condenser is of copper -3 mm. thick. The box is 4 X 3 X 1-5 cm., so that it can pass freely into the calorimeter. Heat op Solution. A calorimeter of thin copper 3 5 cms. diameter and 4 cms. high is used to hold the salt. Mechanical Equivalent op Heat. The apparatus used is that of Puluj, with the arrangement for measuring the frictional couple improved. To diminish the correction for cooling, the cross-section of the float has been increased to six or seven times that given on p. 145. APPENDIX 363 SECTIOS XXXI. XXXIIL XXXV. XXXVI. XXXVII. XXXVIII. XXXIX XLI. XLII. FrtEQUENCY BY THE SyREN. It is convenient to apply part of the pressure on the top of the bellows by hand, as this enables the note of the syren to be readily varied by a slight change of the force used. LiSSAJOUS* FlOURES. Large forks of frequencies 128 and 256 are suitable. Sextant. The objects viewed should be at a considerable distance from the observer, and their angular distance apart should not be greater than about 20". Curvature of Lenses. The optic bench is 1 metre long and 14 cms. wide. The lens is an ordinary reading lens of II cms. diameter. Total Reflection. The cube has an edge of 4*5 cms., and the slit is 22 cms. above tiie table. Magnifying powers. A board painted white, with a black .scale painted on it with divisions about 8 cms. apart, and each fifth numlxired, makes a suitable object for the first exercise. Use of Spectroscope. The salts are contained in 2 inch corked sample bottles placed in holes in a block of wood 20 cms. by 7 cms. The crayons are the ordinary coloured crayons sold in sixpenny boxes. The Spectrometer and its Adjustments. The model of the vernier is about '^0 cms. long. The graduations are on paper gunimc<l to wooden blocks and varnished. Refra<tivk Inoe.x of a Solid. The small piece of mirror is provided to enable light to be reflected down on to the verniers when the readings are l)eing taken. 364 APPENDIX SECTION XLIV. XLVI. XLVII. XLVIII. XLIX. Photometry. The standard sperm candle weighs J of a pound. LII. LIII. Diffraction Grating. One of Thorp's replicas answers the purpose admirably. Rotation of Plane of Polarisation. I. The tubes containing the sugar solution should be of glass or be glass-lined, to prevent the acid used in inverting the sugar acting on them. Rotation of Plane of Polarisation. II. There are other means of producing the half shadow field, as effective as the plate of quartz described on p. 227. Magnetic Fields. The horizontal graduated scales are about 3 mms. thick and 80 or 100 cms. long. A deflecting magnet about 6 X -5 X '6 cms. is suitable. Magnetic Survey. The apparatus consists of a circular scale of 15 — 20 cms. diameter, over which a magnetic needle 4 cms. long with a cross section of 04 — '06 sq. cm., is suspended by a line untwisted silk fibre about 20 cms. long. The needle should have a light aluminium pointer attached to it. The scale and needle are enclosed in a box open at the top. Comparison of Current Meters. The resistance coils are of No. 16 platinoid wire, and are arranged parallel to each other between terminals which may be connected together in any desired way by means of strips of copper. Application op Ohm's Law, The wires are stretched along graduated metre scales screwed to rods of wood 1 '5 cms. thick and 6 cms. broad. APPENDIX 365 SECTION LIV. LV. LVI. LVIII. UX. LX. LXII. LXIIl. LXIV. Arrangement of Cells. The outer jars of the cells are 13 cms. high and 10 cms. diameter outside, and the porous pots 12 cms. high and 5*5 cms. diameter. Water Voltameter. The voltameter of Fig. 105 is due to Kohlrausch and may be obtained from Messrs Hartmann and Braun, Bockenheim near Frankfort o. M. Copper Voltameter. The copper plates used are about 6 cms. square, and each has a lug about 3 cms. long, which fits into and makes good electrical contact with a spring clip. Measurement of Resistance. Coils of alK)ut 4, 20, 100, 500, and 2000 ohms give sufficient practice to the student. High Resistance. Strips of insulating material clamped between binding screws answer for the last portion of the exercise. Low Resistance. A coil of about 30 cms. of No. 14 platinoid wire, with connexions made to it at several points of its length, is convenient. Resistance of a Cell. A condenser of J to 1 microfarad capacity u suitable. Carey Foster's Method. The two equal coils may conveniently be made of bare platinoid or manganin wire wound on glass and kept in the same bath of petroleum. Temperature Change of Resistance. About 4 metres of No. 38 copper, and 2 metres of No. 29 platinoid or manganin wire wound on strips of insulating fibre, form suitable coils. 366 APPENDIX SECTION LXV. LXVI. LXVIII. LXX. LXXI. LXXII. LXXIII. Resistance of Electrolytes. A tube about 16 cms. long and 4 -5 cms. internal diameter enables all the solutions to be tested by means of one Post OflSce box of 1 1 ,000 ohms. Clark Cell. The vertical limbs of the H tube are 8 cms. long and 1 -6 cms. diameter. Potentiometer Method of Measuring Currents. The wire used for the standard low resistance should be capable of carrying the current without the increase of temperature produced altering its resistance appreciably. Equivalent of Heat by the Electrical Method. The heating coil consists of 90 cms. of No. 14 platinoid wire. Induction of Electric Currents. The 3 inner coils each consist of one layer of 50 turns of No. 20 copper wire, and are wound in parallel. The 3 outer coils are similar but have 40 turns. Short lengths of platinoid wire are inter- posed between the ends of the coils and the terminals, so that whether one coil, or two in series, or three in series are in circuit the resistance is the same. Inductance of a Coil. The coil consists of 575 turns of No. 20 copper wire, the mean radius of a turn being 25-3 cms. Leakage and Absorption of Condensers. The dielectric of the condenser used consists of paraffined paper, one side of which is shellac varnished. INDEX. Accurate weighing, 40 Alcohol, refractive index of, 210 Angles, optical methods of measuring, 154 Balance, 28 ,, sensibility of, 33 Bifilar suspension, 71 Biprism, 216 Boiling-point depending on pressure, 112 Buoyancy correction, 41, 48 Calibration of graduated tube, 22 „ spirit level, 17 Capacity of condensers, 357 Clark cell, 319 Compound pendulum, 75 Condensers, 350 „ capacity of, 357 „ comparison of, 354 „ leakage of, 351 „ residual charge, 352 Cooling correction in calorimetry, 126 „ laws of, 119 „ Newton's law, 123 Copper voltameter, 267 Current meters, comparison of, 248 ,, ,, standardisation of, 267 Currents measured by potentiometer, 827 Cniratures of lenses, 157 Density of a liquid, 62 „ ,, quartz, 65, 58 „ ,, a solid, 53 Dew point, 116 Diffraction grating, 220 Discarding of anneoessary deoinuds, 181 Earth's horizontal force, 284 Bleeiro^ynamometer, 246 Electrolytes, resistance of, 816 Electromotive forces, comparison of» 323 Equivalent simple pendulum, 76 Equivalent simple pendulum, deter- mination of, 79 Error, fractional, 6 „ probable, 3 Errors of observation, 1 Expansion of a gas at constant prea> sure, 108 „ M liquid, 100 solid, 97 Foster's method of comparing resis- tances, 307 Frequency of tuning fork, 146 Galvanometer, adjustment of, 272 ,, constant by water elec- trolysis, 265 „ dead beat, 282 ,, how to increase sen- sitiveness, 279 ,, resistance of, 800 ,, tangent adjustment of, 245 Gauss's eyepiece, 200 „ method of weighing, 45 Heat of solution, 140 Humidity, relative, 116 Hydrometer, 65 Hygrometer, Daniell's, 116 India-rubber, specific heat of, 128 Inductance, determination of, 840 Induction, 835 Inertia, moments of, 67, 71 Interference of light, 216 Kelvin current balance, 271 Kundt's method of measuring velocity of sound, 149 Latent beat of steam, 187 „ ,, water, 185 368 INDEX Laurent's polarimeter, 229 Least squares, method of, 7 Length, measurement of, 9 Lenses, curvatures and powers, 167 Level, calibration of, 17 Lissajous' figures, 151 Lodge's modification of Mance*iB method, 304 Logarithmic decrement, 283 Magnetic fields, 232 ,, intensity, 238 Magnifying power, 176 Mance's method, 304 Mechanical equivalent of heat by electrical method, 332 Mechanical equivalent of heat by friction, 142 Modulus of rigidity, 86 Young's, 72 Mohr's balance, 64 Moments of inertia, 57, 71 Newton's law of cooling, 123 Ohm's law, verification of, 253 Personal equation, 8 Photometry, 213 Platinum thermometers, 313 Polarisation, rotation of plane, 224, 226 Post Office box, 286 Potentiometer, 325 ,, method of measuring currents, 327 Pressure, effect on boiling point, 112 ,, coefficient of gas, 104 Prism, determination of angle, 205 Puluj's friction cones, 142 Quartz, specific heat of, 132 Refractive index, 173 „ „ of liquids, 209 of solids, 202 „ power, 212 Resistance of a cell, 304 „ of electrolytes, 315 ; brass, expansion of, 99 „ of a galvanometer, 300 Resistance, high, measurement of, 293 ,, low, measurement of, 298 ,, measurement of, 285, 307 „ temperature coefficient of, 310 Resistivity, measurement of, 292 Eider of balance, 33 Rigidity, modulus of, 86 Saturation, fractional, 116 Sensibility of balance, 33 Sextant, 162 ,, index error, 165 Specific gravity, 53 „ „ flask, 57 ,, heat of india-rubber, ,, „ quartz, 132 Spectra, mapping of, 184 Spectrometer, adjustment of. Spectroscope, adjustment of, ,, reduction to lengths, 187 Steam, latent heat of, 137 Subdivision by eye, 9 Surface tension, 93 Syren, 147 128 194 181 wave Temperature coefficient of electrical resistance, 310 Temperature coefficient of expansion of gas, 108 Thermo-electricity, 329 Time, intervals of, 12 Total reflection, 173 Tube, calibration of, 22 Velocity of sound, measurement of, 149 Viscosity, 89 Voltaic cells, arrangement of, 258 Voltameter, water, 262 ,, copper, 267 Water, expansion of, 102 ,, latent heat of, 135 „ voltameter, 262 Wave length, measurement of, 220 Wet and dry bulb thermometer, 118 Young's modulus, 72 CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. UNIVERSITY OF TORONTO LIBRARY Acme Library Card Pockvt Inder P»t. " Ref. lodet File." Made hj LIBRABT BIJILEAU